- MathJax
**Volume 1, 1990**- Number 1
- Andrzej Trybulec.
Tarski Grothendieck Set Theory,
Formalized Mathematics 1(1), pages 9-11, 1990. MML Identifier: TARSKI

**Summary**: This is the first part of the axiomatics of the Mizar system. It includes the axioms of the Tarski Grothendieck set theory. They are: the axiom stating that everything is a set, the extensionality axiom, the definitional axiom of the singleton, the definitional axiom of the pair, the definitional axiom of the union of a family of sets, the definitional axiom of the boolean (the power set) of a set, the regularity axiom, the definitional axiom of the ordered pair, the Tarski's axiom~A introduced in \cite{TARSKI:1} (see also \cite{TARSKI:2}), and the Fr\ae nkel scheme. Also, the definition of equinumerosity is introduced.

- Andrzej Trybulec.
Built-in Concepts,
Formalized Mathematics 1(1), pages 13-15, 1990. MML Identifier: AXIOMS

**Summary**: This abstract contains the second part of the axiomatics of the Mizar system (the first part is in abstract \cite{TARSKI.ABS}). The axioms listed here characterize the Mizar built-in concepts that are automatically attached to every Mizar article. We give definitional axioms of the following concepts: element, subset, Cartesian product, domain (non empty subset), subdomain (non empty subset of a domain), set domain (domain consisting of sets). Axioms of strong arithmetics of real numbers are also included.

- Zinaida Trybulec, Halina Swieczkowska.
Boolean Properties of Sets,
Formalized Mathematics 1(1), pages 17-23, 1990. MML Identifier: BOOLE

**Summary**: This article contains proofs of the theorems which are obvious if the directive 'requirements BOOLE;' will be added to enviroment declaration of the Mizar article.

- Andrzej Trybulec.
Enumerated Sets,
Formalized Mathematics 1(1), pages 25-34, 1990. MML Identifier: ENUMSET1

**Summary**: We prove basic facts about enumerated sets: definitional theorems and their immediate consequences, some theorems related to the decomposition of an enumerated set into union of two sets, facts about removing elements that occur more than once, and facts about permutations of enumerated sets (with the length $\le$ 4). The article includes also schemes enabling instantiation of up to nine universal quantifiers.

- Krzysztof Hryniewiecki.
Basic Properties of Real Numbers,
Formalized Mathematics 1(1), pages 35-40, 1990. MML Identifier: REAL_1

**Summary**: Basic facts of arithmetics of real numbers are presented: definitions and properties of the complement element, the inverse element, subtraction and division; some basic properties of the set REAL (e.g. density), and the scheme of separation for sets of reals.

- Grzegorz Bancerek.
The Fundamental Properties of Natural Numbers,
Formalized Mathematics 1(1), pages 41-46, 1990. MML Identifier: NAT_1

**Summary**: Some fundamental properties of addition, multiplication, order relations, exact division, the remainder, divisibility, the least common multiple, the greatest common divisor are presented. A proof of Euclid algorithm is also given.

- Czeslaw Bylinski.
Some Basic Properties of Sets,
Formalized Mathematics 1(1), pages 47-53, 1990. MML Identifier: ZFMISC_1

**Summary**: In this article some basic theorems about singletons, pairs, power sets, unions of families of sets, and the cartesian product of two sets are proved.

- Czeslaw Bylinski.
Functions and Their Basic Properties,
Formalized Mathematics 1(1), pages 55-65, 1990. MML Identifier: FUNCT_1

**Summary**: The definitions of the mode Function and the graph of a function are introduced. The graph of a function is defined to be identical with the function. The following concepts are also defined: the domain of a function, the range of a function, the identity function, the composition of functions, the 1-1 function, the inverse function, the restriction of a function, the image and the inverse image. Certain basic facts about functions and the notions defined in the article are proved.

- Zinaida Trybulec.
Properties of Subsets,
Formalized Mathematics 1(1), pages 67-71, 1990. MML Identifier: SUBSET_1

**Summary**: The text includes theorems concerning properties of subsets, and some operations on sets. The functions yielding improper subsets of a set, i.e. the empty set and the set itself are introduced. Functions and predicates introduced for sets are redefined. Some theorems about enumerated sets are proved.

- Edmund Woronowicz.
Relations and Their Basic Properties,
Formalized Mathematics 1(1), pages 73-83, 1990. MML Identifier: RELAT_1

**Summary**: We define here: mode Relation as a set of pairs, the domain, the codomain, and the field of relation; the empty and the identity relations, the composition of relations, the image and the inverse image of a set under a relation. Two predicates, = and \subseteq, and three functions, $\cup$, $\cap$ and $\setminus$ are redefined. Basic facts about the above mentioned notions are presented.

- Edmund Woronowicz, Anna Zalewska.
Properties of Binary Relations,
Formalized Mathematics 1(1), pages 85-89, 1990. MML Identifier: RELAT_2

**Summary**: The paper contains definitions of some properties of binary relations: reflexivity, irreflexivity, symmetry, asymmetry, antisymmetry, connectedness, strong connectedness, and transitivity. Basic theorems relating the above mentioned notions are given.

- Grzegorz Bancerek.
The Ordinal Numbers,
Formalized Mathematics 1(1), pages 91-96, 1990. MML Identifier: ORDINAL1

**Summary**: In the beginning of the article we show some consequences of the regularity axiom. In the second part we introduce the successor of a set and the notions of transitivity and connectedness wrt membership relation. Then we define ordinal numbers as transitive and connected sets, and we prove some theorems of them and of their sets. Lastly we introduce the concept of a transfinite sequence and we show transfinite induction and schemes of defining by transfinite induction.

- Andrzej Trybulec.
Tuples, Projections and Cartesian Products,
Formalized Mathematics 1(1), pages 97-105, 1990. MML Identifier: MCART_1

**Summary**: The purpose of this article is to define projections of ordered pairs, and to introduce triples and quadruples, and their projections. The theorems in this paper may be roughly divided into two groups: theorems describing basic properties of introduced concepts and theorems related to the regularity, analogous to those proved for ordered pairs by Cz. Byli\'nski \cite{ZFMISC_1.ABS}. Cartesian products of subsets are redefined as subsets of Cartesian products.

- Grzegorz Bancerek, Krzysztof Hryniewiecki.
Segments of Natural Numbers and Finite Sequences,
Formalized Mathematics 1(1), pages 107-114, 1990. MML Identifier: FINSEQ_1

**Summary**: We define the notion of an initial segment of natural numbers and prove a number of their properties. Using this notion we introduce finite sequences, subsequences, the empty sequence, a sequence of a domain, and the operation of concatenation of two sequences.

- Andrzej Trybulec.
Domains and Their Cartesian Products,
Formalized Mathematics 1(1), pages 115-122, 1990. MML Identifier: DOMAIN_1

**Summary**: The article includes: theorems related to domains, theorems related to Cartesian products presented earlier in various articles and simplified here by substituting domains for sets and omitting the assumption that the sets involved must not be empty. Several schemes and theorems related to Fraenkel operator are given. We also redefine subset yielding functions such as the pair of elements of a set and the union of two subsets of a set.

- Grzegorz Bancerek.
The Well Ordering Relations,
Formalized Mathematics 1(1), pages 123-129, 1990. MML Identifier: WELLORD1

**Summary**: Some theorems about well ordering relations are proved. The goal of the article is to prove that every two well ordering relations are either isomorphic or one of them is isomorphic to a segment of the other. The following concepts are defined: the segment of a relation induced by an element, well founded relations, well ordering relations, the restriction of a relation to a set, and the isomorphism of two relations. A number of simple facts is presented.

- Grzegorz Bancerek.
A Model of ZF Set Theory Language,
Formalized Mathematics 1(1), pages 131-145, 1990. MML Identifier: ZF_LANG

**Summary**: The goal of this article is to construct a language of the ZF set theory and to develop a notational and conceptual base which facilitates a convenient usage of the language.

- Beata Padlewska.
Families of Sets,
Formalized Mathematics 1(1), pages 147-152, 1990. MML Identifier: SETFAM_1

**Summary**: The article contains definitions of the following concepts: family of sets, family of subsets of a set, the intersection of a family of sets. Functors $\cup$, $\cap$, and $\setminus$ are redefined for families of subsets of a set. Some properties of these notions are presented.

- Czeslaw Bylinski.
Functions from a Set to a Set,
Formalized Mathematics 1(1), pages 153-164, 1990. MML Identifier: FUNCT_2

**Summary**: The article is a continuation of \cite{FUNCT_1.ABS}. We define the following concepts: a function from a set $X$ into a set $Y$, denoted by ``Function of $X$,$Y$'', the set of all functions from a set $X$ into a set $Y$, denoted by Funcs($X$,$Y$), and the permutation of a set (mode Permutation of $X$, where $X$ is a set). Theorems and schemes included in the article are reformulations of the theorems of \cite{FUNCT_1.ABS} in the new terminology. Also some basic facts about functions of two variables are proved.

- Agata Darmochwal.
Finite Sets,
Formalized Mathematics 1(1), pages 165-167, 1990. MML Identifier: FINSET_1

**Summary**: The article contains the definition of a finite set based on the notion of finite sequence. Some theorems about properties of finite sets and finite families of sets are proved.

- Czeslaw Bylinski.
Graphs of Functions,
Formalized Mathematics 1(1), pages 169-173, 1990. MML Identifier: GRFUNC_1

**Summary**: The graph of a function is defined in \cite{FUNCT_1.ABS}. In this paper the graph of a function is redefined as a Relation. Operations on functions are interpreted as the corresponding operations on relations. Some theorems about graphs of functions are proved.

- Czeslaw Bylinski.
Binary Operations,
Formalized Mathematics 1(1), pages 175-180, 1990. MML Identifier: BINOP_1

**Summary**: In this paper we define binary and unary operations on domains. We also define the following predicates concerning the operations: $\dots$ is commutative, $\dots$ is associative, $\dots$ is the unity of $\dots$, and $\dots$ is distributive wrt $\dots$. A number of schemes useful in justifying the existence of the operations are proved.

- Edmund Woronowicz.
Relations Defined on Sets,
Formalized Mathematics 1(1), pages 181-186, 1990. MML Identifier: RELSET_1

**Summary**: The article includes theorems concerning properties of relations defined as a subset of the Cartesian product of two sets (mode Relation of $X$,$Y$ where $X$,$Y$ are sets). Some notions, introduced in \cite{RELAT_1.ABS} such as domain, codomain, field of a relation, composition of relations, image and inverse image of a set under a relation are redefined.

- Andrzej Trybulec, Agata Darmochwal.
Boolean Domains,
Formalized Mathematics 1(1), pages 187-190, 1990. MML Identifier: FINSUB_1

**Summary**: BOOLE DOMAIN is a SET DOMAIN that is closed under union and difference. This condition is equivalent to being closed under symmetric difference and one of the following operations: union, intersection or difference. We introduce the set of all finite subsets of a set $A$, denoted by Fin $A$. The mode Finite Subset of a set $A$ is introduced with the mother type: Element of Fin $A$. In consequence, ``Finite Subset of \dots '' is an elementary type, therefore one may use such types as ``set of Finite Subset of $A$'', ``[(Finite Subset of $A$), Finite Subset of $A$]'', and so on. The article begins with some auxiliary theorems that belong really to \cite{BOOLE.ABS} or \cite{ORDINAL1.ABS} but are missing there. Moreover, bool $A$ is redefined as a SET DOMAIN, for an arbitrary set $A$.

- Grzegorz Bancerek.
Models and Satisfiability,
Formalized Mathematics 1(1), pages 191-199, 1990. MML Identifier: ZF_MODEL

**Summary**: The article includes schemes of defining by structural induction, and definitions and theorems related to: the set of variables which have free occurrences in a ZF-formula, the set of all valuations of variables in a model, the set of all valuations which satisfy a ZF-formula in a model, the satisfiability of a ZF-formula in a model by a valuation, the validity of a ZF-formula in a model, the axioms of ZF-language, the model of the ZF set theory.

- Grzegorz Bancerek.
The Contraction Lemma,
Formalized Mathematics 1(1), pages 201-203, 1990. MML Identifier: ZF_COLLA

**Summary**: The article includes the proof of the contraction lemma which claims that every class in which the axiom of extensionality is valid is isomorphic with a transitive class. In this article the isomorphism (wrt membership relation) of two sets is defined. It is based on \cite{MOST:1}.

- Wojciech A. Trybulec.
Axioms of Incidency,
Formalized Mathematics 1(1), pages 205-213, 1990. MML Identifier: INCSP_1

**Summary**: This article is based on {\it ``Foundations of Geometry''} by Karol Borsuk and Wanda Szmielew (\cite{BORSUK:1}). The fourth axiom of incidency is weakened. In \cite{BORSUK:1} it has the form {\it for any plane there exist three non-collinear points in the plane} and in the article {\it for any plane there exists one point in the plane}. The original axiom is proved. The article includes: theorems concerning collinearity of points and coplanarity of points and lines, basic theorems concerning lines and planes, fundamental existence theorems, theorems concerning intersection of lines and planes.

- Stanislaw Zukowski.
Introduction to Lattice Theory,
Formalized Mathematics 1(1), pages 215-222, 1990. MML Identifier: LATTICES

**Summary**: A lattice is defined as an algebra on a nonempty set with binary operations join and meet which are commutative and associative, and satisfy the absorption identities. The following kinds of lattices are considered: distributive, modular, bounded (with zero and unit elements), complemented, and Boolean (with complement). The article includes also theorems which immediately follow from definitions.

- Beata Padlewska, Agata Darmochwal.
Topological Spaces and Continuous Functions,
Formalized Mathematics 1(1), pages 223-230, 1990. MML Identifier: PRE_TOPC

**Summary**: The paper contains a definition of topological space. The following notions are defined: point of topological space, subset of topological space, subspace of topological space, and continuous function.

- Miroslaw Wysocki, Agata Darmochwal.
Subsets of Topological Spaces,
Formalized Mathematics 1(1), pages 231-237, 1990. MML Identifier: TOPS_1

**Summary**: The article contains some theorems about open and closed sets. The following topological operations on sets are defined: closure, interior and frontier. The following notions are introduced: dense set, boundary set, nowheredense set and set being domain (closed domain and open domain), and some basic facts concerning them are proved.

- Beata Padlewska.
Connected Spaces,
Formalized Mathematics 1(1), pages 239-244, 1990. MML Identifier: CONNSP_1

**Summary**: The following notions are defined: separated sets, connected spaces, connected sets, components of a topological space, the component of a point. The definition of the boundary of a set is also included. The singleton of a point of a topological space is redefined as a subset of the space. Some theorems about these notions are proved.

- Czeslaw Bylinski.
Basic Functions and Operations on Functions,
Formalized Mathematics 1(1), pages 245-254, 1990. MML Identifier: FUNCT_3

**Summary**: We define the following mappings: the characteristic function of a subset of a set, the inclusion function (injection or embedding), the projections from a Cartesian product onto its arguments and diagonal function (inclusion of a set into its Cartesian square). Some operations on functions are also defined: the products of two functions (the complex function and the more general product-function), the function induced on power sets by the image and inverse-image. Some simple propositions related to the introduced notions are proved.

- Andrzej Trybulec.
Tarski Grothendieck Set Theory,
Formalized Mathematics 1(1), pages 9-11, 1990. MML Identifier: TARSKI
- Number 2
- Agata Darmochwal.
Families of Subsets, Subspaces and Mappings in Topological Spaces,
Formalized Mathematics 1(2), pages 257-261, 1990. MML Identifier: TOPS_2

**Summary**: This article is a continuation of \cite{TOPS_1.ABS}. Some basic theorems about families of sets in a topological space have been proved. Following redefinitions have been made: singleton of a set as a family in the topological space and results of boolean operations on families as a family of the topological space. Notion of a family of complements of sets and a closed (open) family have been also introduced. Next some theorems refer to subspaces in a topological space: some facts about types in a subspace, theorems about open and closed sets and families in a subspace. A notion of restriction of a family has been also introduced and basic properties of this notion have been proved. The last part of the article is about mappings. There are proved necessary and sufficient conditions for a mapping to be continuous. A notion of homeomorphism has been defined next. Theorems about homeomorphisms of topological spaces have been also proved.

- Jan Popiolek.
Some Properties of Functions Modul and Signum,
Formalized Mathematics 1(2), pages 263-264, 1990. MML Identifier: ANAL_1

**Summary**:

- Jan Popiolek.
Some Properties of Functions Modul and Signum,
Formalized Mathematics 1(2), pages 263-264, 1990. MML Identifier: ABSVALUE

**Summary**: The article includes definitions and theorems concerning basic properties of the following functions: $|x|$ -- modul of real number, sgn $x$ -- signum of real number.

- Grzegorz Bancerek.
Zermelo Theorem and Axiom of Choice,
Formalized Mathematics 1(2), pages 265-267, 1990. MML Identifier: WELLORD2

**Summary**: The article is continuation of \cite{WELLORD1.ABS} and \cite{ORDINAL1.ABS}, and the goal of it is show that Zermelo theorem (every set has a relation which well orders it - proposition (26)) and axiom of choice (for every non-empty family of non-empty and separate sets there is set which has exactly one common element with arbitrary family member - proposition (27)) are true. It is result of the Tarski's axiom A introduced in \cite{TARSKI:1} and repeated in \cite{TARSKI.ABS}. Inclusion as a settheoretical binary relation is introduced, the correspondence of well ordering relations to ordinal numbers is shown, and basic properties of equinumerosity are presented. Some facts are based on \cite{KURAT-MOST:1}.

- Jaroslaw Kotowicz.
Real Sequences and Basic Operations on Them,
Formalized Mathematics 1(2), pages 269-272, 1990. MML Identifier: SEQ_1

**Summary**: Definition of real sequence and operations on sequences (multiplication of sequences and multiplication by a real number, addition, subtraction, division and absolute value of sequence) are given.

- Jaroslaw Kotowicz.
Convergent Sequences and the Limit of Sequences,
Formalized Mathematics 1(2), pages 273-275, 1990. MML Identifier: SEQ_2

**Summary**: The article contains definitions and same basic properties of bounded sequences (above and below), convergent sequences and the limit of sequences. In the article there are some properties of real numbers useful in the other theorems of this article.

- Grzegorz Bancerek.
Properties of ZF Models,
Formalized Mathematics 1(2), pages 277-280, 1990. MML Identifier: ZFMODEL1

**Summary**: The article deals with the concepts of satisfiability of ZF set theory language formulae in a model (a non-empty family of sets) and the axioms of ZF theory introduced in \cite{MOST:1}. It is shown that the transitive model satisfies the axiom of extensionality and that it satisfies the axiom of pairs if and only if it is closed to pair operation; it satisfies the axiom of unions if and only if it is closed to union operation, etc. The conditions which are satisfied by arbitrary model of ZF set theory are also shown. Besides introduced are definable and parametrically definable functions.

- Grzegorz Bancerek.
Sequences of Ordinal Numbers,
Formalized Mathematics 1(2), pages 281-290, 1990. MML Identifier: ORDINAL2

**Summary**: In the first part of the article we introduce the following operations: On $X$ that yields the set of all ordinals which belong to the set $X$, Lim $X$ that yields the set of all limit ordinals which belong to $X$, and inf $X$ and sup $X$ that yield the minimal ordinal belonging to $X$ and the minimal ordinal greater than all ordinals belonging to $X$, respectively. The second part of the article starts with schemes that can be used to justify the correctness of definitions based on the transfinite induction (see \cite{ORDINAL1.ABS} or \cite{KURAT-MOST:1}). The schemes are used to define addition, product and power of ordinal numbers. The operations of limes inferior and limes superior of sequences of ordinals are defined and the concepts of limit of ordinal sequence and increasing and continuous sequence are introduced.

- Wojciech A. Trybulec.
Vectors in Real Linear Space,
Formalized Mathematics 1(2), pages 291-296, 1990. MML Identifier: RLVECT_1

**Summary**: In this article we introduce a notion of real linear space, operations on vectors: addition, multiplication by real number, inverse vector, subtraction. The sum of finite sequence of the vectors is also defined. Theorems that belong rather to \cite{NAT_1.ABS} or \cite{FINSEQ_1.ABS} are proved.

- Wojciech A. Trybulec.
Subspaces and Cosets of Subspaces in Real Linear Space,
Formalized Mathematics 1(2), pages 297-301, 1990. MML Identifier: RLSUB_1

**Summary**: The following notions are introduced in the article: subspace of a real linear space, zero subspace and improper subspace, coset of a subspace. The relation of a subset of the vectors being linearly closed is also introduced. Basic theorems concerning those notions are proved in the article.

- Piotr Rudnicki, Andrzej Trybulec.
A First Order Language,
Formalized Mathematics 1(2), pages 303-311, 1990. MML Identifier: QC_LANG1

**Summary**: In the paper a first order language is constructed. It includes the universal quantifier and the following propositional connectives: truth, negation, and conjunction. The variables are divided into three kinds: bound variables, fixed variables, and free variables. An infinite number of predicates for each arity is provided. Schemes of structural induction and schemes justifying definitions by structural induction have been proved. The concept of a closed formula (a formula without free occurrences of bound variables) is introduced.

- Wojciech A. Trybulec.
Partially Ordered Sets,
Formalized Mathematics 1(2), pages 313-319, 1990. MML Identifier: ORDERS_1

**Summary**: In the beginning of this article we define the choice function of a non-empty set family that does not contain $\emptyset$ as introduced in \cite[pages 88--89]{KURAT:1}. We define order of a set as a relation being reflexive, antisymmetric and transitive in the set, partially ordered set as structure non-empty set and order of the set, chains, lower and upper cone of a subset, initial segments of element and subset of partially ordered set. Some theorems that belong rather to \cite{ZFMISC_1.ABS} or \cite{RELAT_2.ABS} are proved.

- Krzysztof Hryniewiecki.
Recursive Definitions,
Formalized Mathematics 1(2), pages 321-328, 1990. MML Identifier: RECDEF_1

**Summary**: The text contains some schemes which allow elimination of definitions by recursion.

- Andrzej Trybulec.
Binary Operations Applied to Functions,
Formalized Mathematics 1(2), pages 329-334, 1990. MML Identifier: FUNCOP_1

**Summary**: In the article we introduce functors yielding to a binary operation its composition with an arbitrary functions on its left side, its right side or both. We prove theorems describing the basic properties of these functors. We introduce also constant functions and converse of a function. The recent concept is defined for an arbitrary function, however is meaningful in the case of functions which range is a subset of a Cartesian product of two sets. Then the converse of a function has the same domain as the function itself and assigns to an element of the domain the mirror image of the ordered pair assigned by the function. In the case of functions defined on a non-empty set we redefine the above mentioned functors and prove simplified versions of theorems proved in the general case. We prove also theorems stating relationships between introduced concepts and such properties of binary operations as commutativity or associativity.

- Eugeniusz Kusak, Wojciech Leonczuk, Michal Muzalewski.
Abelian Groups, Fields and Vector Spaces,
Formalized Mathematics 1(2), pages 335-342, 1990. MML Identifier: VECTSP_1

**Summary**: This text includes definitions of the Abelian group, field and vector space over a field and some elementary theorems about them.

- Eugeniusz Kusak, Wojciech Leonczuk, Michal Muzalewski.
Parallelity Spaces,
Formalized Mathematics 1(2), pages 343-348, 1990. MML Identifier: PARSP_1

**Summary**: In the monography \cite{SZMIELEW:1} W. Szmielew introduced the parallelity planes $\langle S$; $\parallel \rangle$, where $\parallel \subseteq S\times S\times S\times S$. In this text we omit upper bound axiom which must be satisfied by the parallelity planes (see also E.Kusak \cite{KUSAK:1}). Further we will list those theorems which remain true when we pass from the parallelity planes to the parallelity spaces. We construct a model of the parallelity space in Abelian group $\langle F\times F\times F; +_F, -_F, {\bf 0}_F \rangle$, where $F$ is a field.

- Eugeniusz Kusak, Wojciech Leonczuk, Michal Muzalewski.
Construction of a bilinear antisymmetric form in simplectic vector space,
Formalized Mathematics 1(2), pages 349-352, 1990. MML Identifier: SYMSP_1

**Summary**: In this text we will present unpublished results by Eu\-ge\-niusz Ku\-sak. It contains an axiomatic description of the class of all spaces $\langle V$; $\perp_\xi \rangle$, where $V$ is a vector space over a field F, $\xi: V \times V \to F$ is a bilinear antisymmetric form i.e. $\xi(x,y) = -\xi(y,x)$ and $x \perp_\xi y $ iff $\xi(x,y) = 0$ for $x$, $y \in V$. It also contains an effective construction of bilinear antisymmetric form $\xi$ for given symplectic space $\langle V$; $\perp \rangle$ such that $\perp = \perp_\xi$. The basic tool used in this method is the notion of orthogonal projection J$(a,b,x)$ for $a,b,x \in V$. We should stress the fact that axioms of orthogonal and symplectic spaces differ only by one axiom, namely: $x\perp y+\varepsilon z \>\&\> y\perp z+\varepsilon x \Rightarrow z\perp x+\varepsilon y. $ For $\varepsilon=+1$ we get the axiom characterizing symplectic geometry. For $\varepsilon=-1$ we get the axiom on three perpendiculars characterizing orthogonal geometry - see \cite{ORTSP_1.ABS}.

- Eugeniusz Kusak, Wojciech Leonczuk, Michal Muzalewski.
Construction of a bilinear symmetric form in orthogonal vector space,
Formalized Mathematics 1(2), pages 353-356, 1990. MML Identifier: ORTSP_1

**Summary**: In this text we present unpublished results by Eu\-ge\-niusz Ku\-sak and Wojciech Leo\'nczuk. They contain an axiomatic description of the class of all spaces $\langle V$; $\perp_\xi \rangle$, where $V$ is a vector space over a field F, $\xi: V \times V \to F$ is a bilinear symmetric form i.e. $\xi(x,y) = \xi(y,x)$ and $x \perp_\xi y$ iff $\xi(x,y) = 0$ for $x$, $y \in V$. They also contain an effective construction of bilinear symmetric form $\xi$ for given orthogonal space $\langle V$; $\perp \rangle$ such that $\perp = \perp_\xi$. The basic tool used in this method is the notion of orthogonal projection J$(a,b,x)$ for $a,b,x \in V$. We should stress the fact that axioms of orthogonal and symplectic spaces differ only by one axiom, namely: $x\perp y+\varepsilon z \>\&\> y\perp z+\varepsilon x \Rightarrow z\perp x+\varepsilon y.$ For $\varepsilon=-1$ we get the axiom on three perpendiculars characterizing orthogonal geometry. For $\varepsilon=+1$ we get the axiom characterizing symplectic geometry - see \cite{SYMSP_1.ABS}.

- Czeslaw Bylinski.
Partial Functions,
Formalized Mathematics 1(2), pages 357-367, 1990. MML Identifier: PARTFUN1

**Summary**: In the article we define partial functions. We also define the following notions related to partial functions and functions themselves: the empty function, the restriction of a function to a partial function from a set into a set, the set of all partial functions from a set into a set, the total functions, the relation of tolerance of two functions and the set of all total functions which are tolerated by a partial function. Some simple propositions related to the introduced notions are proved. In the beginning of this article we prove some auxiliary theorems and schemes related to the articles: \cite{FUNCT_1.ABS} and \cite{FUNCT_2.ABS}.

- Andrzej Trybulec.
Semilattice Operations on Finite Subsets,
Formalized Mathematics 1(2), pages 369-376, 1990. MML Identifier: SETWISEO

**Summary**: In the article we deal with a binary operation that is associative, commutative. We define for such an operation a functor that depends on two more arguments: a finite set of indices and a function indexing elements of the domain of the operation and yields the result of applying the operation to all indexed elements. The definition has a restriction that requires that either the set of indices is non empty or the operation has the unity. We prove theorems describing some properties of the functor introduced. Most of them we prove in two versions depending on which requirement is fulfilled. In the second part we deal with the union of finite sets that enjoys mentioned above properties. We prove analogs of the theorems proved in the first part. We precede the main part of the article with auxiliary theorems related to boolean properties of sets, enumerated sets, finite subsets, and functions. We define a casting function that yields to a set the empty set typed as a finite subset of the set. We prove also two schemes of the induction on finite sets.

- Grzegorz Bancerek.
Cardinal Numbers,
Formalized Mathematics 1(2), pages 377-382, 1990. MML Identifier: CARD_1

**Summary**: We present the choice function rule in the beginning of the article. In the main part of the article we formalize the base of cardinal theory. In the first section we introduce the concept of cardinal numbers and order relations between them. We present here Cantor-Bernstein theorem and other properties of order relation of cardinals. In the second section we show that every set has cardinal number equipotence to it. We introduce notion of alephs and we deal with the concept of finite set. At the end of the article we show two schemes of cardinal induction. Some definitions are based on \cite{GUZ-ZBIER:1} and \cite{KURAT-MOST:1}.

- Agata Darmochwal.
Compact Spaces,
Formalized Mathematics 1(2), pages 383-386, 1990. MML Identifier: COMPTS_1

**Summary**: The article contains definition of a compact space and some theorems about compact spaces. The notions of a cover of a set and a centered family are defined in the article to be used in these theorems. A set is compact in the topological space if and only if every open cover of the set has a finite subcover. This definition is equivalent, what has been shown next, to the following definition: a set is compact if and only if a subspace generated by that set is compact. Some theorems about mappings and homeomorphisms of compact spaces have been also proved. The following schemes used in proofs of theorems have been proved in the article: FuncExChoice -- the scheme of choice of a function, BiFuncEx -- the scheme of parallel choice of two functions and the theorem about choice of a finite counter image of a finite image.

- Wojciech A. Trybulec, Grzegorz Bancerek.
Kuratowski -- Zorn Lemma,
Formalized Mathematics 1(2), pages 387-393, 1990. MML Identifier: ORDERS_2

**Summary**: The goal of this article is to prove Kuratowski - Zorn lemma. We prove it in a number of forms (theorems and schemes). We introduce the following notions: a relation is a quasi (or partial, or linear) order, a relation quasi (or partially, or linearly) orders a set, minimal and maximal element in a relation, inferior and superior element of a relation, a set has lower (or upper) Zorn property w.r.t. a relation. We prove basic theorems concerning those notions and theorems that relate them to the notions introduced in \cite{ORDERS_1.ABS}. At the end of the article we prove some theorems that belong rather to \cite{RELAT_1.ABS}, \cite{RELAT_2.ABS} or \cite{WELLORD1.ABS}.

- Wojciech A. Trybulec.
Operations on Subspaces in Real Linear Space,
Formalized Mathematics 1(2), pages 395-399, 1990. MML Identifier: RLSUB_2

**Summary**: In this article the following operations on subspaces of real linear space are intoduced: sum, intersection and direct sum. Some theorems about those notions are proved. We define linear complement of a subspace. Some theorems about decomposition of a vector onto two subspaces and onto subspace and its linear complement are proved. We also show that a set of subspaces with operations sum and intersection is a lattice. At the end of the article theorems that belong rather to \cite{BOOLE.ABS}, \cite{RLVECT_1.ABS}, \cite{RLSUB_1.ABS} or \cite{LATTICES.ABS} are proved.

- Andrzej Ndzusiak.
$\sigma$-Fields and Probability,
Formalized Mathematics 1(2), pages 401-407, 1990. MML Identifier: PROB_1

**Summary**: This article contains definitions and theorems concerning basic properties of following objects: - a field of subsets of given nonempty set; - a sequence of subsets of given nonempty set; - a $\sigma$-field of subsets of given nonempty set and events from this $\sigma$-field; - a probability i.e. $\sigma$-additive normed measure defined on previously introduced $\sigma$-field; - a $\sigma$-field generated by family of subsets of given set; - family of Borel Sets.

- Czeslaw Bylinski.
Introduction to Categories and Functors,
Formalized Mathematics 1(2), pages 409-420, 1990. MML Identifier: CAT_1

**Summary**: The category is introduced as an ordered 5-tuple of the form $\langle O, M, dom, cod, \cdot, id \rangle$ where $O$ (objects) and $M$ (morphisms) are arbitrary nonempty sets, $dom$ and $cod$ map $M$ onto $O$ and assign to a morphism domain and codomain, $\cdot$ is a partial binary map from $M \times M$ to $M$ (composition of morphisms), $id$ applied to an object yields the identity morphism. We define the basic notions of the category theory such as hom, monic, epi, invertible. We next define functors, the composition of functors, faithfulness and fullness of functors, isomorphism between categories and the identity functor.

- Grzegorz Bancerek.
Introduction to Trees,
Formalized Mathematics 1(2), pages 421-427, 1990. MML Identifier: TREES_1

**Summary**: The article consists of two parts: the first one deals with the concept of the prefixes of a finite sequence, the second one introduces and deals with the concept of tree. Besides some auxiliary propositions concerning finite sequences are presented. The trees are introduced as non-empty sets of finite sequences of natural numbers which are closed on prefixes and on sequences of less numbers (i.e. if $\langle n_1$, $n_2$, $\dots$, $n_k\rangle$ is a vertex (element) of a tree and $m_i \leq n_i$ for $i = 1$, $2$, $\dots$, $k$, then $\langle m_1$, $m_2$, $\dots$, $m_k\rangle$ also is). Finite trees, elementary trees with $n$ leaves, the leaves and the subtrees of a tree, the inserting of a tree into another tree, with a node used for determining the place of insertion, antichains of prefixes, and height and width of finite trees are introduced.

- Agata Darmochwal.
Families of Subsets, Subspaces and Mappings in Topological Spaces,
Formalized Mathematics 1(2), pages 257-261, 1990. MML Identifier: TOPS_2
- Number 3
- Bogdan Nowak, Slawomir Bialecki.
Zermelo's Theorem,
Formalized Mathematics 1(3), pages 431-432, 1990. MML Identifier: WELLSET1

**Summary**: The article contains direct proof of Zermelo's theorem about the existence of a well ordering for any set and the lemma the proof depends on.

- Jozef Bialas.
Group and Field Definitions,
Formalized Mathematics 1(3), pages 433-439, 1990. MML Identifier: REALSET1

**Summary**: The article contains exactly the same definitions of group and field as those in \cite{DIEUDONNE}. These definitions were prepared without the help of the definitions and properties of {\it Nat} and {\it Real} modes included in the MML. This is the first of a series of articles in which we are going to introduce the concept of the set of real numbers in a elementary axiomatic way.

- Konrad Raczkowski, Pawel Sadowski.
Equivalence Relations and Classes of Abstraction,
Formalized Mathematics 1(3), pages 441-444, 1990. MML Identifier: EQREL_1

**Summary**: In this article we deal with the notion of equivalence relation. The main properties of equivalence relations are proved. Then we define the classes of abstraction determined by an equivalence relation. Finally, the connections between a partition of a set and an equivalence relation are presented. We introduce the following notation of modes: {\it Equivalence Relation, a partition}.

- Andrzej Trybulec, Czeslaw Bylinski.
Some Properties of Real Numbers,
Formalized Mathematics 1(3), pages 445-449, 1990. MML Identifier: SQUARE_1

**Summary**: We define the following operations on real numbers: $max(x,y)$, $min(x,y)$, $x^2$, $\sqrt{x}$. We prove basic properties of introduced operations. A number of auxiliary theorems absent in \cite{REAL_1.ABS} and \cite{ABSVALUE.ABS} is proved.

- Grzegorz Bancerek.
Connectives and Subformulae of the First Order Language,
Formalized Mathematics 1(3), pages 451-458, 1990. MML Identifier: QC_LANG2

**Summary**: In the article the development of the first order language defined in \cite{QC_LANG1.ABS} is continued. The following connectives are introduced: implication ($\Rightarrow$), disjunction ($\vee$), and equivalence ($\Leftrightarrow$). We introduce also the existential quantifier ($\exists$) and FALSUM. Some theorems on disjunctive, conditional, biconditional and existential formulae are proved and their selector functors are introduced. The second part of the article deals with notions of subformula, proper subformula and immediate constituent of a QC-formula.

- Czeslaw Bylinski, Grzegorz Bancerek.
Variables in Formulae of the First Order Language,
Formalized Mathematics 1(3), pages 459-469, 1990. MML Identifier: QC_LANG3

**Summary**: We develop the first order language defined in \cite{QC_LANG1.ABS}. We continue the work done in the article \cite{QC_LANG2.ABS}. We prove some schemes of defining by structural induction. We deal with notions of closed subformulae and of still not bound variables in a formula. We introduce the concept of the set of all free variables and the set of all fixed variables occurring in a formula.

- Jaroslaw Kotowicz.
Monotone Real Sequences. Subsequences,
Formalized Mathematics 1(3), pages 471-475, 1990. MML Identifier: SEQM_3

**Summary**: The article contains definitions of constant, increasing, decreasing, non decreasing, non increasing sequences, the definition of a subsequence and their basic properties.

- Jaroslaw Kotowicz.
Convergent Real Sequences. Upper and Lower Bound of Sets of Real Numbers,
Formalized Mathematics 1(3), pages 477-481, 1990. MML Identifier: SEQ_4

**Summary**: The article contains theorems about convergent sequences and the limit of sequences occurring in \cite{SEQ_2.ABS} such as Bolzano-Weierstrass theorem, Cauchy theorem and others. Bounded sets of real numbers and lower and upper bound of subset of real numbers are defined.

- Michal Muzalewski.
Midpoint algebras,
Formalized Mathematics 1(3), pages 483-488, 1990. MML Identifier: MIDSP_1

**Summary**: In this article basic properties of midpoint algebras are proved. We define a congruence relation $\equiv$ on bound vectors and free vectors as the equivalence classes of $\equiv$.

- Pawel Sadowski, Andrzej Trybulec, Konrad Raczkowski.
The Fundamental Logic Structure in Quantum Mechanics,
Formalized Mathematics 1(3), pages 489-494, 1990. MML Identifier: QMAX_1

**Summary**: In this article we present the logical structure given by four axioms of Mackey \cite{MACKEY} in the set of propositions of Quantum Mechanics. The equivalence relation (PropRel(Q)) in the set of propositions (Prop Q) for given Quantum Mechanics Q is considered. The main text for this article is \cite{EQREL_1.ABS} where the structure of quotient space and the properties of equivalence relations, classes and partitions are studied.

- Andrzej Trybulec.
Function Domains and Fr\aenkel Operator,
Formalized Mathematics 1(3), pages 495-500, 1990. MML Identifier: FRAENKEL

**Summary**: We deal with a non--empty set of functions and a non--empty set of functions from a set $A$ to a non--empty set $B$. In the case when $B$ is a non--empty set, $B^A$ is redefined. It yields a non--empty set of functions from $A$ to $B$. An element of such a set is redefined as a function from $A$ to $B$. Some theorems concerning these concepts are proved, as well as a number of schemes dealing with infinity and the Axiom of Choice. The article contains a number of schemes allowing for simple logical transformations related to terms constructed with the Fr{\ae}nkel Operator.

- Michal J. Trybulec.
Integers,
Formalized Mathematics 1(3), pages 501-505, 1990. MML Identifier: INT_1

**Summary**: In the article the following concepts were introduced: the set of integers (${\Bbb Z }$) and its elements (integers), congruences ($i_1 \equiv i_2 (\mathop{\rm mod} i_3)$), the ceiling and floor functors ($\mathopen{\lceil} x \mathclose{\rceil}$ and $\mathopen{\lfloor} x \mathclose{\rfloor}$), also the fraction part of a real number (frac), the integer division ($\div$) and remainder of integer division (mod). The following schemes were also included: the separation scheme ({\it SepInt}), the schemes of integer induction ({\it Int\_Ind\_Down}, {\it Int\_Ind\_Up}, {\it Int\_Ind\_Full}), the minimum ({\it Int\_Min}) and maximum ({\it Int\_Max}) schemes (the existence of minimum and maximum integers enjoying a given property).

- Czeslaw Bylinski.
The Complex Numbers,
Formalized Mathematics 1(3), pages 507-513, 1990. MML Identifier: COMPLEX1

**Summary**: We define the set $\Bbb C$ of complex numbers as the set of all ordered pairs $z =\langle a,b\rangle$ where $a$ and $b$ are real numbers and where addition and multiplication are defined. We define the real and imaginary parts of $z$ and denote this by $a = \Re(z)$, $b = \Im(z)$. These definitions satisfy all the axioms for a field. $0_{\Bbb C} = 0+0i$ and $1_{\Bbb C} = 1+0i$ are identities for addition and multiplication respectively, and there are multiplicative inverses for each non zero element in $\Bbb C$. The difference and division of complex numbers are also defined. We do not interpret the set of all real numbers $\Bbb R$ as a subset of $\Bbb C$. From here on we do not abandon the ordered pair notation for complex numbers. For example: $i^2 = (0+1i)^2 = -1+0i \neq -1$. We conclude this article by introducing two operations on $\Bbb C$ which are not field operations. We define the absolute value of $z$ denoted by $|z|$ and the conjugate of $z$ denoted by $z^\ast$.

- Grzegorz Bancerek.
Ordinal Arithmetics,
Formalized Mathematics 1(3), pages 515-519, 1990. MML Identifier: ORDINAL3

**Summary**: At the beginning the article contains some auxiliary theorems concerning the constructors defined in papers \cite{ORDINAL1.ABS} and \cite{ORDINAL2.ABS}. Next simple properties of addition and multiplication of ordinals are shown, e.g. associativity of addition. Addition and multiplication of a transfinite sequence of ordinals and a ordinal are also introduced here. The goal of the article is the proof that the distributivity of multiplication wrt addition and the associativity of multiplication hold. Additionally new binary functors of ordinals are introduced: subtraction, exact division, and remainder and some of their basic properties are presented.

- Czeslaw Bylinski.
The Modification of a Function by a Function and the Iteration of the Composition of a Function,
Formalized Mathematics 1(3), pages 521-527, 1990. MML Identifier: FUNCT_4

**Summary**: In the article we introduce some operations on functions. We define the natural ordering relation on functions. The fact that a function $f$ is less than a function $g$ we denote by $f \leq g$ and we define by $\hbox{graph} f \subseteq \hbox{graph} f$. In the sequel we define the modifications of a function $f$ by a function $g$ denoted $f \hbox{+$\cdot$} g$ and the $n$-th iteration of the composition of a function $f$ denoted by $f^n$. We prove some propositions related to the introduced notions.

- Czeslaw Bylinski.
Finite Sequences and Tuples of Elements of a Non-empty Sets,
Formalized Mathematics 1(3), pages 529-536, 1990. MML Identifier: FINSEQ_2

**Summary**: The first part of the article is a continuation of \cite{FINSEQ_1.ABS}. Next, we define the identity sequence of natural numbers and the constant sequences. The main part of this article is the definition of tuples. The element of a set of all sequences of the length $n$ of $D$ is called a tuple of a non-empty set $D$ and it is denoted by element of $D^{n}$. Also some basic facts about tuples of a non-empty set are proved.

- Grzegorz Bancerek.
Curried and Uncurried Functions,
Formalized Mathematics 1(3), pages 537-541, 1990. MML Identifier: FUNCT_5

**Summary**: In the article following functors are introduced: the projections of subsets of the Cartesian product, the functor which for every function $f:X \times Y \to Z$ gives some curried function ($X \to(Y \to Z)$), and the functor which from curried functions makes uncurried functions. Some of their properties and some properties of the set of all functions from a set into a set are also shown.

- Grzegorz Bancerek.
Cardinal Arithmetics,
Formalized Mathematics 1(3), pages 543-547, 1990. MML Identifier: CARD_2

**Summary**: In the article addition, multiplication and power operation of cardinals are introduced. Presented are some properties of equipotence of Cartesian products, basic cardinal arithmetics laws (transformativity, associativity, distributivity), and some facts about finite sets.

- Eugeniusz Kusak, Wojciech Leonczuk.
Fano-Desargues Parallelity Spaces,
Formalized Mathematics 1(3), pages 549-553, 1990. MML Identifier: PARSP_2

**Summary**: This article is the second part of Parallelity Space. It contains definition of a Fano-Desargues space, axioms of a Fano-Desargues parallelity space, definition of the relations: collinearity, parallelogram and directed congruence and some basic facts concerned with them.

- Henryk Oryszczyszyn , Krzysztof Prazmowski.
Real Functions Spaces,
Formalized Mathematics 1(3), pages 555-561, 1990. MML Identifier: FUNCSDOM

**Summary**: This abstract contains a construction of the domain of functions defined in an arbitrary nonempty set, with values being real numbers. In every such set of functions we introduce several algebraic operations, which yield in this set the structures of a real linear space, of a ring, and of a real algebra. Formal definitions of such concepts are given.

- Grzegorz Bancerek.
Tarski's Classes and Ranks,
Formalized Mathematics 1(3), pages 563-567, 1990. MML Identifier: CLASSES1

**Summary**: In the article the Tarski's classes (non-empty families of sets satisfying Tarski's axiom A given in \cite{TARSKI.ABS}) and the rank sets are introduced and some of their properties are shown. The transitive closure and the rank of a set is given here too.

- Wojciech A. Trybulec.
Non-contiguous Substrings and One-to-one Finite Sequences,
Formalized Mathematics 1(3), pages 569-573, 1990. MML Identifier: FINSEQ_3

**Summary**: This text is a continuation of \cite{FINSEQ_1.ABS}. We prove a number of theorems concerning both notions introduced there and one-to-one finite sequences. We introduce a function that removes from a string elements of the string that belongs to a given set.

- Wojciech A. Trybulec.
Pigeon Hole Principle,
Formalized Mathematics 1(3), pages 575-579, 1990. MML Identifier: FINSEQ_4

**Summary**: We introduce the notion of a predicate that states that a function is one-to-one at a given element of its domain (i.e. counterimage of image of the element is equal to its singleton). We also introduce some rather technical functors concerning finite sequences: the lowest index of the given element of the range of the finite sequence, the substring preceding (and succeeding) the first occurrence of given element of the range. At the end of the article we prove the pigeon hole principle.

- Wojciech A. Trybulec.
Linear Combinations in Real Linear Space,
Formalized Mathematics 1(3), pages 581-588, 1990. MML Identifier: RLVECT_2

**Summary**: The article is continuation of \cite{RLVECT_1.ABS}. At the beginning we prove some theorems concerning sums of finite sequence of vectors. We introduce the following notions: sum of finite subset of vectors, linear combination, carrier of linear combination, linear combination of elements of a given set of vectors, sum of linear combination. We also show that the set of linear combinations is a real linear space. At the end of article we prove some auxiliary theorems that should be proved in \cite{BOOLE.ABS}, \cite{FUNCT_1.ABS}, \cite{FINSET_1.ABS}, \cite{NAT_1.ABS} or \cite{REAL_1.ABS}.

- Grzegorz Bancerek.
K\"onig's Theorem,
Formalized Mathematics 1(3), pages 589-593, 1990. MML Identifier: CARD_3

**Summary**: In the article the sum and product of any number of cardinals are introduced and their relationships to addition, multiplication and to other concepts are shown. Then the K\"onig's theorem is proved. The theorem that the cardinal of union of increasing family of sets of power less than some cardinal {\bf m} is not greater than {\bf m}, is given too.

- Bogdan Nowak, Grzegorz Bancerek.
Universal Classes,
Formalized Mathematics 1(3), pages 595-600, 1990. MML Identifier: CLASSES2

**Summary**: In the article we have shown that there exist universal classes, i.e. there are sets which are closed w.r.t. basic set theory operations.

- Henryk Oryszczyszyn, Krzysztof Prazmowski.
Analytical Ordered Affine Spaces,
Formalized Mathematics 1(3), pages 601-605, 1990. MML Identifier: ANALOAF

**Summary**: In the article with a given arbitrary real linear space we correlate the (ordered) affine space defined in terms of a directed parallelity of segments. The abstract contains a construction of the ordered affine structure associated with a vector space; this is a structure of the type which frequently occurs in geometry and consists of the set of points and a binary relation on segments. For suitable underlying vector spaces we prove that the corresponding affine structures are ordered affine spaces or ordered affine planes, i.e. that they satisfy appropriate axioms. A formal definition of an arbitrary ordered affine space and an arbitrary ordered affine plane is given.

- Stanislawa Kanas, Adam Lecko, Mariusz Startek.
Metric Spaces,
Formalized Mathematics 1(3), pages 607-610, 1990. MML Identifier: METRIC_1

**Summary**: In this paper we define the metric spaces. Two examples of metric spaces are given. We define the discrete metric and the metric on the real axis. Moreover the open ball, the close ball and the sphere in metric spaces are introduced. We also prove some theorems concerning these concepts.

- Henryk Oryszczyszyn, Krzysztof Prazmowski.
Ordered Affine Spaces Defined in Terms of Directed Parallelity -- Part I,
Formalized Mathematics 1(3), pages 611-615, 1990. MML Identifier: DIRAF

**Summary**: In the article we consider several geometrical relations in given arbitrary ordered affine space defined in terms of directed parallelity. In particular we introduce the notions of the nondirected parallelity of segments, of collinearity, and the betweenness relation determined by the given relation of directed parallelity. The obtained structures satisfy commonly accepted axioms for affine spaces. At the end of the article we introduce a formal definition of affine space and affine plane (defined in terms of parallelity of segments).

- Henryk Oryszczyszyn, Krzysztof Prazmowski.
Parallelity and Lines in Affine Spaces,
Formalized Mathematics 1(3), pages 617-621, 1990. MML Identifier: AFF_1

**Summary**: In the article we introduce basic notions concerning affine spaces and investigate their fundamental properties. We define the function which to every nondegenerate pair of points assigns the line joining them and we extend the relation of parallelity to a relation between segments and lines, and between lines.

- Bogdan Nowak, Slawomir Bialecki.
Zermelo's Theorem,
Formalized Mathematics 1(3), pages 431-432, 1990. MML Identifier: WELLSET1
- Number 4
- Henryk Oryszczyszyn, Krzysztof Prazmowski.
Classical Configurations in Affine Planes,
Formalized Mathematics 1(4), pages 625-633, 1990. MML Identifier: AFF_2

**Summary**: The classical sequence of implications which hold between Desargues and Pappus Axioms is proved. Formally Minor and Major Desargues Axiom (as suitable properties -- predicates -- of an affine plane) together with all its indirect forms are introduced; the same procedure is applied to Pappus Axioms. The so called Trapezium Desargues Axiom is also considered.

- Eugeniusz Kusak, Henryk Oryszczyszyn, Krzysztof Prazmowski.
Affine Localizations of Desargues Axiom,
Formalized Mathematics 1(4), pages 635-642, 1990. MML Identifier: AFF_3

**Summary**: Several affine localizations of Major Desargues Axiom together with its indirect forms are introduced. Logical relationships between these formulas and between them and the classical Desargues Axiom are demonstrated.

- Czeslaw Bylinski.
Binary Operations Applied to Finite Sequences,
Formalized Mathematics 1(4), pages 643-649, 1990. MML Identifier: FINSEQOP

**Summary**: The article contains some propositions and theorems related to \cite{FUNCOP_1.ABS} and \cite{FINSEQ_2.ABS}. The notions introduced in \cite{FUNCOP_1.ABS} are extended to finite sequences. A number of additional propositions related to this notions are proved. There are also proved some properties of distributive operations and unary operations. The notation and propositions for inverses are introduced.

- Czeslaw Bylinski.
Semigroup operations on finite subsets,
Formalized Mathematics 1(4), pages 651-656, 1990. MML Identifier: SETWOP_2

**Summary**: A continuation of \cite{SETWISEO.ABS}. The propositions and theorems proved in \cite{SETWISEO.ABS} are extended to finite sequences. Several additional theorems related to semigroup operations of functions not included in \cite{SETWISEO.ABS} are proved. The special notation for operations on finite sequences is introduced.

- Wojciech Skaba.
The Collinearity Structure,
Formalized Mathematics 1(4), pages 657-659, 1990. MML Identifier: COLLSP

**Summary**: The text includes basic axioms and theorems concerning the collinearity structure based on Wanda Szmielew \cite{SZMIELEW:1}, pp. 18--20. Collinearity is defined as a relation on Cartesian product $\mizleftcart S, S, S \mizrightcart$ of set $S$. The basic text is preceeded with a few auxiliary theorems (e.g: ternary relation). Then come the two basic axioms of the collinearity structure: A1.1.1 and A1.1.2 and a few theorems. Another axiom: Aks dim, which states that there exist at least 3 non-collinear points, excludes the trivial structures (i.e. pairs $\llangle S, \mizleftcart S, S, S \mizrightcart\rrangle$). Following it the notion of a line is included and several additional theorems are appended.

- Czeslaw Bylinski.
The Sum and Product of Finite Sequences of Real Numbers,
Formalized Mathematics 1(4), pages 661-668, 1990. MML Identifier: RVSUM_1

**Summary**: Some operations on the set of $n$-tuples of real numbers are introduced. Addition, difference of such $n$-tuples, complement of a $n$-tuple and multiplication of these by real numbers are defined. In these definitions more general properties of binary operations applied to finite sequences from \cite{FINSEQOP.ABS} are used. Then the fact that certain properties are satisfied by those operations is demonstrated directly from \cite{FINSEQOP.ABS}. Moreover some properties can be recognized as being those of real vector space. Multiplication of $n$-tuples of real numbers and square power of $n$-tuple of real numbers using for notation of some properties of finite sums and products of real numbers are defined, followed by definitions of the finite sum and product of $n$-tuples of real numbers using notions and properties introduced in \cite{SETWOP_2.ABS}. A number of propositions and theorems on sum and product of finite sequences of real numbers are proved. As additional properties there are proved some properties of real numbers and set representations of binary operations on real numbers.

- Czeslaw Bylinski.
A Classical First Order Language,
Formalized Mathematics 1(4), pages 669-676, 1990. MML Identifier: CQC_LANG

**Summary**: The aim is to construct a language for the classical predicate calculus. The language is defined as a subset of the language constructed in \cite{QC_LANG1.ABS}. Well-formed formulas of this language are defined and some usual connectives and quantifiers of \cite{QC_LANG1.ABS}, \cite{QC_LANG2.ABS} are accordingly. We prove inductive and definitional schemes for formulas of our language. Substitution for individual variables in formulas of the introduced language is defined. This definition is borrowed from \cite{POGORZELSKI.1975}. For such purpose some auxiliary notation and propositions are introduced.

- Henryk Oryszczyszyn, Krzysztof Prazmowski, Malgorzata Prazmowska.
Classical and Non--classical Pasch Configurations in Ordered Affine Planes,
Formalized Mathematics 1(4), pages 677-680, 1990. MML Identifier: PASCH

**Summary**: Several configuration axioms, which are commonly called in the literature ``Pasch Axioms'' are introduced; three of them were investigated by Szmielew and concern invariantability of the betweenness relation under parallel projections, and two other were introduced by Tarski. It is demonstrated that they all are consequences of the trapezium axiom, adopted to characterize ordered affine spaces.

- Marek Chmur.
The Lattice of Real Numbers. The Lattice of Real Functions,
Formalized Mathematics 1(4), pages 681-684, 1990. MML Identifier: REAL_LAT

**Summary**: A proof of the fact, that $\llangle {\Bbb R}, {\rm max}, {\rm min} \rrangle$ is a lattice (real lattice). Some basic properties (real lattice is distributive and modular) of it are proved. The same is done for the set ${\Bbb R}^A$ with operations: max($f(A)$) and min($f(A)$), where ${\Bbb R}^A$ means the set of all functions from $A$ (being non-empty set) to $\Bbb R$, $f$ is just such a function.

- Grzegorz Lewandowski, Krzysztof Prazmowski.
A Construction of an Abstract Space of Congruence of Vectors,
Formalized Mathematics 1(4), pages 685-688, 1990. MML Identifier: TDGROUP

**Summary**: In the class of abelian groups a subclass of two-divisible-groups is singled out, and in the latter, a subclass of uniquely-two-divisible-groups. With every such a group a special geometrical structure, more precisely the structure of ``congruence of vectors'' is correlated. The notion of ``affine vector space'' (denoted by AffVect) is introduced. This term is defined by means of suitable axiom system. It is proved that every structure of the congruence of vectors determined by a non trivial uniquely two divisible group is a affine vector space.

- Agata Darmochwal.
A First--Order Predicate Calculus,
Formalized Mathematics 1(4), pages 689-695, 1990. MML Identifier: CQC_THE1

**Summary**: A continuation of \cite{CQC_LANG.ABS}, with an axiom system of first-order predicate theory. The consequence Cn of a set of formulas $X$ is defined as the intersection of all theories containing $X$ and some basic properties of it has been proved (monotonicity, idempotency, completeness etc.). The notion of a proof of given formula is also introduced and it is shown that ${\rm Cn} X = \{~p: p $ has a proof w.r.t. $ X\}$. First 14 theorems are rather simple facts. I just wanted them to be included in the data base.

- Jaroslaw Kotowicz.
Partial Functions from a Domain to a Domain,
Formalized Mathematics 1(4), pages 697-702, 1990. MML Identifier: PARTFUN2

**Summary**: The value of a partial function from a domain to a domain and a inverse partial function are introduced. The value and inverse function were defined in the article \cite{FUNCT_1.ABS}, but new definitions are introduced. The basic properties of the value, the inverse partial function, the identity partial function, the composition of partial functions, the $1{-}1$ partial function, the restriction of a partial function, the image, the inverse image and the graph are proved. Constant partial functions are introduced, too.

- Jaroslaw Kotowicz.
Partial Functions from a Domain to the Set of Real Numbers,
Formalized Mathematics 1(4), pages 703-709, 1990. MML Identifier: RFUNCT_1

**Summary**: Basic operations in the set of partial functions which map a domain to the set of all real numbers are introduced. They include adition, subtraction, multiplication, division, multipication by a real number and also module. Main properties of these operations are proved. A definition of the partial function bounded on a set (bounded below and bounded above) is presented. There are theorems showing the laws of conservation of totality and boundedness for operations of partial functions. The characteristic function of a subset of a domain as a partial function is redefined and a few properties are proved.

- Grzegorz Bancerek.
Increasing and Continuous Ordinal Sequences,
Formalized Mathematics 1(4), pages 711-714, 1990. MML Identifier: ORDINAL4

**Summary**: Concatenation of two ordinal sequences, the mode of all ordinals belonging to a universe and the mode of sequences of them with length equal to the rank of the universe are introduced. Besides, the increasing and continuous transfinite sequences, the limes of ordinal sequences and the power of ordinals, and the fact that every increasing and continuous transfinite sequence has critical numbers (fixed points) are discussed.

- Henryk Oryszczyszyn, Krzysztof Prazmowski.
Transformations in Affine Spaces,
Formalized Mathematics 1(4), pages 715-723, 1990. MML Identifier: TRANSGEO

**Summary**: Two classes of bijections of its point universe are correlated with every affine structure. The first class consists of the transformations, called formal isometries, which map every segment onto congruent segment, the second class consists of the automorphisms of such a structure. Each of these two classes of bijections forms a group for a given affine structure, if it satisfies a very weak axiom system (models of these axioms are called congruence spaces); formal isometries form a normal subgroup in the group of automorphism. In particular ordered affine spaces and affine spaces are congruence spaces; therefore formal isometries of these structures can be considered. They are called positive dilatations and dilatations, resp. For convenience the class of negative dilatations, transformations which map every ``vector" onto parallel ``vector", but with opposite sense, is singled out. The class of translations is distinguished as well. Basic facts concerning all these types of transformations are established, like rigidity, decomposition principle, introductory group-theoretical properties. At the end collineations of affine spaces and their properties are investigated; for affine planes it is proved that the class of collineations coincides with the class of bijections preserving lines.

- Czeslaw Bylinski.
Subcategories and Products of Categories,
Formalized Mathematics 1(4), pages 725-732, 1990. MML Identifier: CAT_2

**Summary**: The {\it subcategory} of a category and product of categories is defined. The {\it inclusion functor} is the injection (inclusion) map $E \atop \hookrightarrow$ which sends each object and each arrow of a Subcategory $E$ of a category $C$ to itself (in $C$). The inclusion functor is faithful. {\it Full subcategories} of $C$, that is, those subcategories $E$ of $C$ such that $\hbox{Hom}_E(a,b) = \hbox{Hom}_C(b,b)$ for any objects $a,b$ of $E$, are defined. A subcategory $E$ of $C$ is full when the inclusion functor $E \atop \hookrightarrow$ is full. The proposition that a full subcategory is determined by giving the set of objects of a category is proved. The product of two categories $B$ and $C$ is constructed in the usual way. Moreover, some simple facts on $bifunctors$ (functors from a product category) are proved. The final notions in this article are that of projection functors and product of two functors ({\it complex} functors and {\it product} functors).

- Edmund Woronowicz.
Many--Argument Relations,
Formalized Mathematics 1(4), pages 733-737, 1990. MML Identifier: MARGREL1

**Summary**: Definitions of relations based on finite sequences. The arity of relation, the set of logical values {\it Boolean} consisting of {\it false} and {\it true} and the operations of negation and conjunction on them are defined.

- Edmund Woronowicz.
Interpretation and Satisfiability in the First Order Logic,
Formalized Mathematics 1(4), pages 739-743, 1990. MML Identifier: VALUAT_1

**Summary**: The main notion discussed is satisfiability. Interpretation and some auxiliary concepts are also introduced.

- Andrzej Ndzusiak.
Probability,
Formalized Mathematics 1(4), pages 745-749, 1990. MML Identifier: PROB_2

**Summary**: Some further theorems concerning probability, among them the equivalent definition of probability are discussed, followed by notions of independence of events and conditional probability and basic theorems on them.

- Henryk Oryszczyszyn, Krzysztof Prazmowski.
Translations in Affine Planes,
Formalized Mathematics 1(4), pages 751-753, 1990. MML Identifier: TRANSLAC

**Summary**: Connections between Minor Desargues Axiom and the transitivity of translation groups are investigated. A formal proof of the theorem which establishes the equivalence of these two properties of affine planes is given. We also prove that, under additional requirement, the plane in question satisfies Fano Axiom; its translation group is uniquely two-divisible.

- Jan Popiolek.
Introduction to Probability,
Formalized Mathematics 1(4), pages 755-760, 1990. MML Identifier: RPR_1

**Summary**: Definitions of Elementary Event and Event in any sample space $E$ are given. Next, the probability of an Event when $E$ is finite is introduced and some properties of this function are investigated. Last part of the paper is devoted to the conditional probability and essential properties of this function (Bayes Theorem).

- Wojciech Leonczuk, Krzysztof Prazmowski.
A Construction of Analytical Projective Space,
Formalized Mathematics 1(4), pages 761-766, 1990. MML Identifier: ANPROJ_1

**Summary**: The collinearity structure denoted by Projec\-ti\-ve\-Spa\-ce(V) is correlated with a given vector space $V$ (over the field of Reals). It is a formalization of the standard construction of a projective space, where points are interpreted as equivalence classes of the relation of proportionality considered in the set of all non-zero vectors. Then the relation of collinearity corresponds to the relation of linear dependence of vectors. Several facts concerning vectors are proved, which correspond in this language to some classical axioms of projective geometry.

- Wojciech Leonczuk, Krzysztof Prazmowski.
Projective Spaces -- Part I,
Formalized Mathematics 1(4), pages 767-776, 1990. MML Identifier: ANPROJ_2

**Summary**: In the class of all collinearity structures a subclass of (dimension free) projective spaces, defined by means of a suitable axiom system, is singled out. Whenever a real vector space V is at least 3-dimensional, the structure ProjectiveSpace(V) is a projective space in the above meaning. Some narrower classes of projective spaces are defined: Fano projective spaces, projective planes, and Fano projective planes. For any of the above classes an explicit axiom system is given, as well as an analytical example. There is also a construction a of 3-dimensional and a 4-dimensional real vector space; these are needed to show appropriate examples of projective spaces.

- Konrad Raczkowski, Pawel Sadowski.
Topological Properties of Subsets in Real Numbers,
Formalized Mathematics 1(4), pages 777-780, 1990. MML Identifier: RCOMP_1

**Summary**: The following notions for real subsets are defined: open set, closed set, compact set, intervals and neighbourhoods. In the sequel some theorems involving above mentioned notions are proved.

- Jaroslaw Kotowicz.
Properties of Real Functions,
Formalized Mathematics 1(4), pages 781-786, 1990. MML Identifier: RFUNCT_2

**Summary**: The list of theorems concerning properties of real sequences and functions is enlarged. (See e.g. \cite{SEQ_1.ABS}, \cite{SEQ_4.ABS}, \cite{RFUNCT_1.ABS}). The monotone real functions are introduced and their properties are discussed.

- Konrad Raczkowski, Pawel Sadowski.
Real Function Continuity,
Formalized Mathematics 1(4), pages 787-791, 1990. MML Identifier: FCONT_1

**Summary**: The continuity of real functions is discussed. There is a function defined on some domain in real numbers which is continuous in a single point and on a subset of domain of the function. Main properties of real continuous functions are proved. Among them there is the Weierstra{\ss} Theorem. Algebraic features for real continuous functions are shown. Lipschitzian functions are introduced. The Lipschitz condition entails continuity.

- Jaroslaw Kotowicz, Konrad Raczkowski.
Real Function Uniform Continuity,
Formalized Mathematics 1(4), pages 793-795, 1990. MML Identifier: FCONT_2

**Summary**: The uniform continuity for real functions is introduced. More theorems concerning continuous functions are given. (See \cite{FCONT_1.ABS}) The Darboux Theorem is exposed. Algebraic features for uniformly continuous functions are presented. Various facts, e.g., a continuous function on a compact set is uniformly continuous are proved.

- Konrad Raczkowski, Pawel Sadowski.
Real Function Differentiability,
Formalized Mathematics 1(4), pages 797-801, 1990. MML Identifier: FDIFF_1

**Summary**: For a real valued function defined on its domain in real numbers the differentiability in a single point and on a subset of the domain is presented. The main elements of differential calculus are developed. The algebraic properties of differential real functions are shown.

- Jaroslaw Kotowicz, Konrad Raczkowski, Pawel Sadowski.
Average Value Theorems for Real Functions of One Variable,
Formalized Mathematics 1(4), pages 803-805, 1990. MML Identifier: ROLLE

**Summary**: Three basic theorems in differential calculus of one variable functions are presented: Rolle Theorem, Lagrange Theorem and Cauchy Theorem. There are also direct conclusions.

- Henryk Oryszczyszyn, Krzysztof Prazmowski.
Classical Configurations in Affine Planes,
Formalized Mathematics 1(4), pages 625-633, 1990. MML Identifier: AFF_2
- Number 5
- Jozef Bialas.
Properties of Fields,
Formalized Mathematics 1(5), pages 807-812, 1990. MML Identifier: REALSET2

**Summary**: The second part of considerations concerning groups and fields. It includes a definition and properties of commutative field $F$ as a structure defined by: the set, a support of $F$, containing two different elements, by two binary operations ${\bf +}_{F}$, ${\bf \cdot}_{F}$ on this set, called addition and multiplication, and by two elements from the support of $F$, ${\bf 0}_{F}$ being neutral for addition and ${\bf 1}_{F}$ being neutral for multiplication. This structure is named a field if $\langle$the support of $F$, ${\bf +}_{F}$, ${\bf 0}_{F} \rangle$ and $\langle$the support of $F$, ${\bf \cdot}_{F}$, ${\bf 1}_{F} \rangle$ are commutative groups and multiplication has the property of left-hand and right-hand distributivity with respect to addition. It is demonstrated that the field $F$ satisfies the definition of a field in the axiomatic approach.

- Grzegorz Bancerek.
Filters -- Part I,
Formalized Mathematics 1(5), pages 813-819, 1990. MML Identifier: FILTER_0

**Summary**: Filters of a lattice, maximal filters (ultrafilters), operation to create a filter generating by an element or by a non-empty subset of the lattice are discussed. Besides, there are introduced implicative lattices such that for every two elements there is an element being pseudo-complement of them. Some facts concerning these concepts are presented too, i.e. for any proper filter there exists an ultrafilter consisting it.

- Wojciech A. Trybulec.
Groups,
Formalized Mathematics 1(5), pages 821-827, 1990. MML Identifier: GROUP_1

**Summary**: Notions of group and abelian group are introduced. The power of an element of a group, order of group and order of an element of a group are defined. Basic theorems concerning those notions are presented.

- Rafal Kwiatek, Grzegorz Zwara.
The Divisibility of Integers and Integer Relatively Primes,
Formalized Mathematics 1(5), pages 829-832, 1990. MML Identifier: INT_2

**Summary**: { We introduce the following notions: 1)

- Michal Muzalewski, Wojciech Skaba.
From Loops to Abelian Multiplicative Groups with Zero,
Formalized Mathematics 1(5), pages 833-840, 1990. MML Identifier: ALGSTR_1

**Summary**: Elementary axioms and theorems on the theory of algebraic structures, taken from the book \cite{SZMIELEW:1}. First a loop structure $\langle G, 0, +\rangle$ is defined and six axioms corresponding to it are given. Group is defined by extending the set of axioms with $(a+b)+c = a+(b+c)$. At the same time an alternate approach to the set of axioms is shown and both sets are proved to yield the same algebraic structure. A trivial example of loop is used to ensure the existence of the modes being constructed. A multiplicative group is contemplated, which is quite similar to the previously defined additive group (called simply a group here), but is supposed to be of greater interest in the future considerations of algebraic structures. The final section brings a slightly more sophisticated structure i.e: a multiplicative loop/group with zero: $\langle G, \cdot, 1, 0\rangle$. Here the proofs are a more challenging and the above trivial example is replaced by a more common (and comprehensive) structure built on the foundation of real numbers.

- Andrzej Kondracki.
Basic Properties of Rational Numbers,
Formalized Mathematics 1(5), pages 841-845, 1990. MML Identifier: RAT_1

**Summary**: A definition of rational numbers and some basic properties of them. Operations of addition, subtraction, multiplication are redefined for rational numbers. Functors numerator (num $p$) and denominator (den $p$) ($p$ is rational) are defined and some properties of them are presented. Density of rational numbers is also given.

- Wojciech A. Trybulec.
Basis of Real Linear Space,
Formalized Mathematics 1(5), pages 847-850, 1990. MML Identifier: RLVECT_3

**Summary**: Notions of linear independence and dependence of set of vectors, the subspace generated by a set of vectors and basis of real linear space are introduced. Some theorems concerning those notions are proved.

- Wojciech A. Trybulec.
Finite Sums of Vectors in Vector Space,
Formalized Mathematics 1(5), pages 851-854, 1990. MML Identifier: VECTSP_3

**Summary**: We define the sum of finite sequences of vectors in vector space. Theorems concerning those sums are proved.

- Wojciech A. Trybulec.
Subgroup and Cosets of Subgroups,
Formalized Mathematics 1(5), pages 855-864, 1990. MML Identifier: GROUP_2

**Summary**: We introduce notion of subgroup, coset of a subgroup, sets of left and right cosets of a subgroup. We define multiplication of two subset of a group, subset of reverse elemens of a group, intersection of two subgroups. We define the notion of an index of a subgroup and prove Lagrange theorem which states that in a finite group the order of the group equals the order of a subgroup multiplied by the index of the subgroup. Some theorems that belong rather to \cite{CARD_1.ABS} are proved.

- Wojciech A. Trybulec.
Subspaces and Cosets of Subspaces in Vector Space,
Formalized Mathematics 1(5), pages 865-870, 1990. MML Identifier: VECTSP_4

**Summary**: We introduce the notions of subspace of vector space and coset of a subspace. We prove a number of theorems concerning those notions. Some theorems that belong rather to \cite{VECTSP_1.ABS} are proved.

- Wojciech A. Trybulec.
Operations on Subspaces in Vector Space,
Formalized Mathematics 1(5), pages 871-876, 1990. MML Identifier: VECTSP_5

**Summary**: Sum, direct sum and intersection of subspaces are introduced. We prove some theorems concerning those notions and the decomposition of vector onto two subspaces. Linear complement of a subspace is also defined. We prove theorems that belong rather to \cite{VECTSP_1.ABS}.

- Wojciech A. Trybulec.
Linear Combinations in Vector Space,
Formalized Mathematics 1(5), pages 877-882, 1990. MML Identifier: VECTSP_6

**Summary**: The notion of linear combination of vectors is introduced as a function from the carrier of a vector space to the carrier of the field. Definition of linear combination of set of vectors is also presented. We define addition and subtraction of combinations and multiplication of combination by element of the field. Sum of finite set of vectors and sum of linear combination is defined. We prove theorems that belong rather to \cite{VECTSP_1.ABS}.

- Wojciech A. Trybulec.
Basis of Vector Space,
Formalized Mathematics 1(5), pages 883-885, 1990. MML Identifier: VECTSP_7

**Summary**: We prove the existence of a basis of a vector space, i.e., a set of vectors that generates the vector space and is linearly independent. We also introduce the notion of a subspace generated by a set of vectors and linear independence of set of vectors.

- Rafal Kwiatek.
Factorial and Newton coefficients,
Formalized Mathematics 1(5), pages 887-890, 1990. MML Identifier: NEWTON

**Summary**: We define the following functions: exponential function (for natural exponent), factorial function and Newton coefficients. We prove some basic properties of notions introduced. There is also a proof of binominal formula. We prove also that $\sum_{k=0}^n {n \choose k}=2^n$.

- Henryk Oryszczyszyn, Krzysztof Prazmowski.
Analytical Metric Affine Spaces and Planes,
Formalized Mathematics 1(5), pages 891-899, 1990. MML Identifier: ANALMETR

**Summary**: We introduce relations of orthogonality of vectors and of orthogonality of segments (considered as pairs of vectors) in real linear space of dimension two. This enables us to show an example of (in fact anisotropic and satisfying theorem on three perpendiculars) metric affine space (and plane as well). These two types of objects are defined formally as "Mizar" modes. They are to be understood as structures consisting of a point universe and two binary relations on segments --- a parallelity relation and orthogonality relation, satisfying appropriate axioms. With every such structure we correlate a structure obtained as a reduct of the given one to the parallelity relation only. Some relationships between metric affine spaces and their affine parts are proved; they enable us to use "affine" facts and constructions in investigating metric affine geometry. We define the notions of line, parallelity of lines and two derived relations of orthogonality: between segments and lines, and between lines. Some basic properties of the introduced notions are proved.

- Wojciech Leonczuk, Krzysztof Prazmowski.
Projective Spaces -- part II,
Formalized Mathematics 1(5), pages 901-907, 1990. MML Identifier: ANPROJ_3

**Summary**:

- Wojciech Leonczuk, Krzysztof Prazmowski.
Projective Spaces -- part III,
Formalized Mathematics 1(5), pages 909-918, 1990. MML Identifier: ANPROJ_4

**Summary**:

- Wojciech Leonczuk, Krzysztof Prazmowski.
Projective Spaces -- part IV,
Formalized Mathematics 1(5), pages 919-927, 1990. MML Identifier: ANPROJ_5

**Summary**:

- Wojciech Leonczuk, Krzysztof Prazmowski.
Projective Spaces -- part V,
Formalized Mathematics 1(5), pages 929-938, 1990. MML Identifier: ANPROJ_6

**Summary**:

- Wojciech Leonczuk, Krzysztof Prazmowski.
Projective Spaces -- part VI,
Formalized Mathematics 1(5), pages 939-947, 1990. MML Identifier: ANPROJ_7

**Summary**:

- Waldemar Korczynski.
Some Elementary Notions of the Theory of Petri Nets,
Formalized Mathematics 1(5), pages 949-953, 1990. MML Identifier: NET_1

**Summary**: Some fundamental notions of the theory of Petri nets are described in Mizar formalism. A Petri net is defined as a triple of the form $\langle {\rm places},\,{\rm transitions},\,{\rm flow} \rangle$ with places and transitions being disjoint sets and flow being a relation included in ${\rm places} \times {\rm transitions}$.

- Wojciech A. Trybulec.
Classes of Conjugation. Normal Subgroups,
Formalized Mathematics 1(5), pages 955-962, 1990. MML Identifier: GROUP_3

**Summary**: Theorems that were not proved in \cite{GROUP_1.ABS} and in \cite{GROUP_2.ABS} are discussed. In the article we define notion of conjugation for elements, subsets and subgroups of a group. We define the classes of conjugation. Normal subgroups of a group and normalizer of a subset of a group or of a subgroup are introduced. We also define the set of all subgroups of a group. An auxiliary theorem that belongs rather to \cite{CARD_2.ABS} is proved.

- Grzegorz Bancerek.
Replacing of Variables in Formulas of ZF Theory,
Formalized Mathematics 1(5), pages 963-972, 1990. MML Identifier: ZF_LANG1

**Summary**: Part one is a supplement to papers \cite{ZF_LANG.ABS}, \cite{ZF_MODEL.ABS}, and \cite{ZFMODEL1.ABS}. It deals with concepts of selector functions, atomic, negative, conjunctive formulas and etc., subformulas, free variables, satisfiability and models (it is shown that axioms of the predicate and the quantifier calculus are satisfied in an arbitrary set). In part two there are introduced notions of variables occurring in a formula and replacing of variables in a formula.

- Grzegorz Bancerek.
The Reflection Theorem,
Formalized Mathematics 1(5), pages 973-977, 1990. MML Identifier: ZF_REFLE

**Summary**: The goal is show that the reflection theorem holds. The theorem is as usual in the Morse-Kelley theory of classes (MK). That theory works with universal class which consists of all sets and every class is a subclass of it. In this paper (and in another Mizar articles) we work in Tarski-Grothendieck (TG) theory (see \cite{TARSKI.ABS}) which ensures the existence of sets that have properties like universal class (i.e. this theory is stronger than MK). The sets are introduced in \cite{CLASSES2.ABS} and some concepts of MK are modeled. The concepts are: the class $On$ of all ordinal numbers belonging to the universe, subclasses, transfinite sequences of non-empty elements of universe, etc. The reflection theorem states that if $A_\xi$ is an increasing and continuous transfinite sequence of non-empty sets and class $A = \bigcup_{\xi \in On} A_\xi$, then for every formula $H$ there is a strictly increasing continuous mapping $F: On \to On$ such that if $\varkappa$ is a critical number of $F$ (i.e. $F(\varkappa) = \varkappa > 0$) and $f \in A_\varkappa^{\bf VAR}$, then $A,f\models H \equiv\ {A_\varkappa},f\models H$. The proof is based on \cite{MOST:1}. Besides, in the article it is shown that every universal class is a model of ZF set theory if $\omega$ (the first infinite ordinal number) belongs to it. Some propositions concerning ordinal numbers and sequences of them are also present.

- Wojciech A. Trybulec.
Binary Operations on Finite Sequences,
Formalized Mathematics 1(5), pages 979-981, 1990. MML Identifier: FINSOP_1

**Summary**: We generalize the semigroup operation on finite sequences introduced in \cite{SETWOP_2.ABS} for binary operations that have a unity or for non-empty finite sequences.

- Andrzej Trybulec.
Finite Join and Finite Meet and Dual Lattices,
Formalized Mathematics 1(5), pages 983-988, 1990. MML Identifier: LATTICE2

**Summary**: The concepts of finite join and finite meet in a lattice are introduced. Some properties of the finite join are proved. After introducing the concept of dual lattice in view of dualism we obtain analogous properties of the meet. We prove these properties of binary operations in a lattice, which are usually included in axioms of the lattice theory. We also introduce the concept of Heyting lattice (a bounded lattice with relative pseudo-complements).

- Grzegorz Bancerek.
Consequences of the Reflection Theorem,
Formalized Mathematics 1(5), pages 989-993, 1990. MML Identifier: ZFREFLE1

**Summary**: Some consequences of the reflection theorem are discussed. To formulate them the notions of elementary equivalence and subsystems, and of models for a set of formulae are introduced. Besides, the concept of cofinality of a ordinal number with second one is used. The consequences of the reflection theorem (it is sometimes called the Scott-Scarpellini lemma) are: (i) If $A_\xi$ is a transfinite sequence as in the reflection theorem (see \cite{ZF_REFLE.ABS}) and $A = \bigcup_{\xi \in On} A_\xi$, then there is an increasing and continuous mapping $\phi$ from $On$ into $On$ such that for every critical number $\kappa$ the set $A_\kappa$ is an elementary subsystem of $A$ ($A_\kappa \prec A$). (ii) There is an increasing continuous mapping $\phi: On \to On$ such that ${\bf R}_\kappa \prec V$ for each of its critical numbers $\kappa$ ($V$ is the universal class and $On$ is the class of all ordinals belonging to $V$). (iii) There are ordinal numbers $\alpha$ cofinal with $\omega$ for which ${\bf R}_\alpha$ are models of ZF set theory. (iv) For each set $X$ from universe $V$ there is a model of ZF $M$ which belongs to $V$ and has $X$ as an element.

- Jozef Bialas.
Properties of Fields,
Formalized Mathematics 1(5), pages 807-812, 1990. MML Identifier: REALSET2
**Volume 2, 1991**- Number 1
- Michal Muzalewski.
Construction of Rings and Left-, Right-, and Bi-Modules over a Ring,
Formalized Mathematics 2(1), pages 3-11, 1991. MML Identifier: VECTSP_2

**Summary**: Definitions of some classes of rings and left-, right-, and bi-modules over a ring and some elementary theorems on rings and skew fields.

- Eugeniusz Kusak.
Desargues Theorem In Projective 3-Space,
Formalized Mathematics 2(1), pages 13-16, 1991. MML Identifier: PROJDES1

**Summary**: Proof of the Desargues theorem in Fanoian projective at least 3-dimensional space.

- Jaroslaw Kotowicz.
The Limit of a Real Function at Infinity,
Formalized Mathematics 2(1), pages 17-28, 1991. MML Identifier: LIMFUNC1

**Summary**:

- Jaroslaw Kotowicz.
One-Side Limits of a Real Function at a Point,
Formalized Mathematics 2(1), pages 29-40, 1991. MML Identifier: LIMFUNC2

**Summary**: We introduce the left-side and the right-side limit of a real function at a point. We prove a few properties of the operations on the proper and improper one-side limits and show that Cauchy and Heine characterizations of the one-side limit are equivalent.

- Wojciech A. Trybulec.
Lattice of Subgroups of a Group. Frattini Subgroup,
Formalized Mathematics 2(1), pages 41-47, 1991. MML Identifier: GROUP_4

**Summary**: We define the notion of a subgroup generated by a set of element of a group and two closely connected notions. Namely lattice of subgroups and Frattini subgroup. The operations in the lattice are the intersection of subgroups (introduced in \cite{GROUP_2.ABS}) and multiplication of subgroups which result is defined as a subgroup generated by a sum of carriers of the two subgroups. In order to define Frattini subgroup and to prove theorems concerning it we introduce notion of maximal subgroup and non-generating element of the group (see \cite[page 30]{KARGAP:1}). Frattini subgroup is defined as in \cite{KARGAP:1} as an intersection of all maximal subgroups. We show that an element of the group belongs to Frattini subgroup of the group if and only if it is a non-generating element. We also prove theorems that should be proved in \cite{NAT_1.ABS} but are not.

- Andrzej Kondracki.
Equalities and Inequalities in Real Numbers,
Formalized Mathematics 2(1), pages 49-63, 1991. MML Identifier: REAL_2

**Summary**: The aim of the article is to give a number of useful theorems concerning equalities and inequalities in real numbers. Some of the theorems are extensions of \cite{REAL_1.ABS} theorems, others were found to be needed in practice.

- Grzegorz Bancerek.
Countable Sets and Hessenberg's Theorem,
Formalized Mathematics 2(1), pages 65-69, 1991. MML Identifier: CARD_4

**Summary**: The concept of countable sets is introduced and there are shown some facts which deal with finite and countable sets. Besides, the article includes theorems and lemmas on the sum and product of infinite cardinals. The most important of them is Hessenberg's theorem which says that for every infinite cardinal {\bf m} the product ${\bf m} \cdot {\bf m}$ is equal to {\bf m}.

- Jaroslaw Kotowicz.
The Limit of a Real Function at a Point,
Formalized Mathematics 2(1), pages 71-80, 1991. MML Identifier: LIMFUNC3

**Summary**: We define the proper and the improper limit of a real function at a point. The main properties of the operations on the limit of function are proved. The connection between the one-side limits and the limit of function at a point are exposed. Equivalent Cauchy and Heine characterizations of the limit of real function at a point are proved.

- Jaroslaw Kotowicz.
The Limit of a Composition of Real Functions,
Formalized Mathematics 2(1), pages 81-92, 1991. MML Identifier: LIMFUNC4

**Summary**: The theorem on the proper and improper limit of a composition of real functions at a point, at infinity and one-side limits at a point are presented.

- Beata Padlewska.
Locally Connected Spaces,
Formalized Mathematics 2(1), pages 93-96, 1991. MML Identifier: CONNSP_2

**Summary**: This article is a continuation of \cite{CONNSP_1.ABS}. We define a neighbourhood of a point and a neighbourhood of a set and prove some facts about them. Then the definitions of a locally connected space and a locally connected set are introduced. Some theorems about locally connected spaces are given (based on \cite{KURAT:1}). We also define a quasi-component of a point and prove some of its basic properties.

- Michal Muzalewski, Leslaw W. Szczerba.
Construction of Finite Sequences over Ring and Left-, Right-, and Bi-Modules over a Ring,
Formalized Mathematics 2(1), pages 97-104, 1991. MML Identifier: ALGSEQ_1

**Summary**: This text includes definitions of finite sequences over rings and left-, right-, and bi-module over a ring treated as functions defined for {\sl all} natural numbers, but with almost everywhere equal to zero. Some elementary theorems are proved, in particular for each category of sequences the scheme about existence is proved. In all four cases, i.e. for rings, left-, right and bi- modules are almost exactly the same, hovewer we do not know how to do the job in Mizar in a different way. The paper is mostly based on the paper \cite{FINSEQ_1.ABS}. In particular the notion of initial segment of natural numbers is introduced which differs from that of \cite{FINSEQ_1.ABS} by starting with zero. This proved to be more convenient for algebraic purposes.

- Krzysztof Hryniewiecki.
Relations of Tolerance,
Formalized Mathematics 2(1), pages 105-109, 1991. MML Identifier: TOLER_1

**Summary**: Introduces notions of relations of tolerance, tolerance set and neighbourhood of an element. The basic properties of relations of tolerance are proved.

- Jan Popiolek.
Real Normed Space,
Formalized Mathematics 2(1), pages 111-115, 1991. MML Identifier: NORMSP_1

**Summary**: We construct a real normed space $\langle V,~\Vert.\Vert\rangle$, where $V$ is a real vector space and $\Vert.\Vert$ is a norm. Auxillary properties of the norm are proved. Next, we introduce a notion of sequence in the real normed space. The basic operations on sequences (addition, subtraction, multiplication by real number) are defined. We study some properties of sequences in the real normed space and the operations on them.

- Jaroslaw Kotowicz.
Schemes of Existence of some Types of Functions,
Formalized Mathematics 2(1), pages 117-123, 1991. MML Identifier: SCHEME1

**Summary**: We prove some useful schemes of existence of real sequences, partial functions from a domain into a domain, partial functions from a set to a set and functions from a domain into a domain. At the beginning we prove some related auxiliary theorems related to the article \cite{NAT_1.ABS}.

- Konrad Raczkowski.
Integer and Rational Exponents,
Formalized Mathematics 2(1), pages 125-130, 1991. MML Identifier: PREPOWER

**Summary**: The article includes definitios and theorems which are needed to define real exponent. The following notions are defined: natural exponent, integer exponent and rational exponent.

- Henryk Oryszczyszyn, Krzysztof Prazmowski.
Homotheties and Shears in Affine Planes,
Formalized Mathematics 2(1), pages 131-133, 1991. MML Identifier: HOMOTHET

**Summary**: We study connections between Major Desargues Axiom and the transitivity of group of homotheties. A formal proof of the theorem which establishes an equivalence of these two properties of affine planes is given. We also study connections between trapezium version of Major Desargues Axiom and the existence of the shears in affine planes. The article contains investigations on ``Scherungssatz".

- Grzegorz Lewandowski, Krzysztof Prazmowski, Bozena Lewandowska.
Directed Geometrical Bundles and Their Analytical Representation,
Formalized Mathematics 2(1), pages 135-141, 1991. MML Identifier: AFVECT0

**Summary**: We introduce the notion of weak directed geometrical bundle. We prove representation theorems for directed and weak directed geometrical bundles which establishes a one-to-one correspondence between such structures and appropriate 2-divisible abelian groups. To this aim we construct over arbitrary weak directed geometrical bundle a group defined entirely in terms of geometrical notions -- the group of (abstract) ``free vectors".

- Grzegorz Bancerek.
Definable Functions,
Formalized Mathematics 2(1), pages 143-145, 1991. MML Identifier: ZFMODEL2

**Summary**: The article is continuation of \cite{ZF_LANG1.ABS} and \cite{ZFMODEL1.ABS}. It deals with concepts of variables occurring in a formula and free variables, replacing of variables in a formula and definable functions. The goal of it is to create a base of facts which are neccesary to show that every model of ZF set theory is a good model, i.e. it is closed with respect to fundamental settheoretical operations (union, intersection, Cartesian product etc.). The base includes the facts concerning with the composition and conditional sum of two definable functions.

- Grzegorz Bancerek, Agata Darmochwal, Andrzej Trybulec.
Propositional Calculus,
Formalized Mathematics 2(1), pages 147-150, 1991. MML Identifier: LUKASI_1

**Summary**: We develop the classical propositional calculus, following \cite{LUKA:1}.

- Czeslaw Bylinski, Andrzej Trybulec.
Complex Spaces,
Formalized Mathematics 2(1), pages 151-158, 1991. MML Identifier: COMPLSP1

**Summary**: We introduce the concept of $n$-dimensional complex space. We prove a number of simple but useful propositions concerning addition, nultiplication by scalars and similar basic concepts. We introduce metric and topology. We prove that $n$-dimensional complex space is a Hausdorff space and that it is regular.

- Jozef Bialas.
Several Properties of Fields. Field Theory,
Formalized Mathematics 2(1), pages 159-162, 1991. MML Identifier: REALSET3

**Summary**: The article includes a continuation of the paper \cite{REALSET2.ABS}. Some simple theorems concerning basic properties of a field are proved.

- Jozef Bialas.
Infimum and Supremum of the Set of Real Numbers. Measure Theory,
Formalized Mathematics 2(1), pages 163-171, 1991. MML Identifier: SUPINF_1

**Summary**: We introduce some properties of the least upper bound and the greatest lower bound of the subdomain of $\overline{\Bbb R}$ numbers, where $\overline{\Bbb R}$ denotes the enlarged set of real numbers, $\overline{\Bbb R} = {\Bbb R} \cup \{-\infty,+\infty\}$. The paper contains definitions of majorant and minorant elements, bounded from above, bounded from below and bounded sets, sup and inf of set, for nonempty subset of $\overline{\Bbb R}$. We prove theorems describing the basic relationships among those definitions. The work is the first part of the series of articles concerning the Lebesgue measure theory.

- Jozef Bialas.
Series of Positive Real Numbers. Measure Theory,
Formalized Mathematics 2(1), pages 173-183, 1991. MML Identifier: SUPINF_2

**Summary**: We introduce properties of a series of nonnegative $\overline{\Bbb R}$ numbers, where $\overline{\Bbb R}$ denotes the enlarged set of real numbers, $\overline{\Bbb R} = {\Bbb R} \cup \{-\infty,+\infty\}$. The paper contains definition of sup $F$ and inf $F$, for $F$ being function, and a definition of a sumable subset of $\overline{\Bbb R}$. We proved the basic theorems regarding the definitions mentioned above. The work is the second part of a series of articles concerning the Lebesgue measure theory.

- Wojciech Skaba, Michal Muzalewski.
From Double Loops to Fields,
Formalized Mathematics 2(1), pages 185-191, 1991. MML Identifier: ALGSTR_2

**Summary**: This paper contains the second part of the set of articles concerning the theory of algebraic structures, based on \cite[pp. 9-12]{SZMIELEW:1} (pages 4--6 of the English edition).\par First the basic structure $\langle F, +, \cdot, 1, 0\rangle$ is defined. Following it the consecutive structures are contemplated in details, including double loop, left quasi-field, right quasi-field, double sided quasi-field, skew field and field. These structures are created by gradually augmenting the basic structure with new axioms of commutativity, associativity, distributivity etc. Each part of the article begins with the set of auxiliary theorems related to the structure under consideration, that are necessary for the existence proof of each defined mode. Next the mode and proof of its existence is included. If the current set of axioms may be replaced with a different and equivalent one, the appropriate proof is performed following the definition of that mode. With the introduction of double loop the ``-'' function is defined. Some interesting features of this function are also included.

- Michal Muzalewski.
Construction of Rings and Left-, Right-, and Bi-Modules over a Ring,
Formalized Mathematics 2(1), pages 3-11, 1991. MML Identifier: VECTSP_2
- Number 2
- Stanislawa Kanas, Jan Stankiewicz.
Metrics in Cartesian Product,
Formalized Mathematics 2(2), pages 193-197, 1991. MML Identifier: METRIC_3

**Summary**: A continuation of paper \cite{METRIC_1.ABS}. It deals with the method of creation of the distance in the Cartesian product of metric spaces. The distance of two points belonging to Cartesian product of metric spaces has been defined as sum of distances of appropriate coordinates (or projections) of these points. It is shown that product of metric spaces with such a distance is a metric space.

- Adam Lecko, Mariusz Startek.
Submetric Spaces -- Part I,
Formalized Mathematics 2(2), pages 199-203, 1991. MML Identifier: SUB_METR

**Summary**:

- Adam Lecko, Mariusz Startek.
On Pseudometric Spaces,
Formalized Mathematics 2(2), pages 205-211, 1991. MML Identifier: METRIC_2

**Summary**:

- Konrad Raczkowski, Andrzej Ndzusiak.
Real Exponents and Logarithms,
Formalized Mathematics 2(2), pages 213-216, 1991. MML Identifier: POWER

**Summary**: Definitions and properties of the following concepts: root, real exponent and logarithm. Also the number $e$ is defined.

- Eugeniusz Kusak, Wojciech Leonczuk.
Hessenberg Theorem,
Formalized Mathematics 2(2), pages 217-219, 1991. MML Identifier: HESSENBE

**Summary**: We prove the Hessenberg theorem which states that every Pappian projective space is Desarguesian.

- Michal Muzalewski, Wojciech Skaba.
Three-Argument Operations and Four-Argument Operations,
Formalized Mathematics 2(2), pages 221-224, 1991. MML Identifier: MULTOP_1

**Summary**: The article contains the definition of three- and four- argument operations. The article introduces also a few operation related schemes: {\it FuncEx3D}, {\it TriOpEx}, {\it Lambda3D}, {\it TriOpLambda}, {\it FuncEx4D}, {\it QuaOpEx}, {\it Lambda4D}, {\it QuaOpLambda}.

- Wojciech Leonczuk, Krzysztof Prazmowski.
Incidence Projective Spaces,
Formalized Mathematics 2(2), pages 225-232, 1991. MML Identifier: INCPROJ

**Summary**: A basis for investigations on incidence projective spaces. With every projective space defined in terms of collinearity relation we associate the incidence structure consisting of points and of lines of the given space. We introduce general notion of projective space defined in terms of incidence and define several properties of such structures (like satisfability of the Desargues Axiom or conditions on the dimension).

- Barbara Konstanta, Urszula Kowieska, Grzegorz Lewandowski, Krzysztof Prazmowski.
One-Dimensional Congruence of Segments, Basic Facts and Midpoint Relation,
Formalized Mathematics 2(2), pages 233-235, 1991. MML Identifier: AFVECT01

**Summary**: We study a theory of one-dimensional congruence of segments. The theory is characterized by a suitable formal axiom system; as a model of this system one can take the structure obtained from any weak directed geometrical bundle, with the congruence interpreted as in the case of ``classical" vectors. Preliminary consequences of our axiom system are proved, basic relations of maximal distance and of midpoint are defined, and several fundamental properties of them are established.

- Andrzej Trybulec.
Algebra of Normal Forms,
Formalized Mathematics 2(2), pages 237-242, 1991. MML Identifier: NORMFORM

**Summary**: We mean by a normal form a finite set of ordered pairs of subsets of a fixed set that fulfils two conditions: elements of it consist of disjoint sets and element of it are incomparable w.r.t. inclusion. The underlying set corresponds to a set of propositional variables but it is arbitrary. The correspondents to a normal form of a formula, e.g. a disjunctive normal form is as follows. The normal form is the set of disjuncts and a disjunct is an ordered pair consisting of the sets of propositional variables that occur in the disjunct non-negated and negated. The requirement that the element of a normal form consists of disjoint sets means that contradictory disjuncts have been removed and the second condition means that the absorption law has been used to shorten the normal form. We construct a lattice $\langle {\Bbb N},\sqcup,\sqcap \rangle$, where $ a \sqcup b = \mu (a \cup b)$ and $a \sqcap b = \mu c$, $c$ being set of all pairs $\langle X_1 \cup Y_1, X_2 \cup Y_2 \rangle$, $\langle X_1, X_2 \rangle \in a$ and $\langle Y_1,Y_2 \rangle \in b$, which consist of disjoiny sets. $\mu a$ denotes here the set of all minimal, w.r.t. inclusion, elements of $a$. We prove that the lattice of normal forms over a set defined in this way is distributive and that $\emptyset$ is the minimal element of it.

- Michal Muzalewski, Leslaw W. Szczerba.
Ordered Rings -- Part I,
Formalized Mathematics 2(2), pages 243-245, 1991. MML Identifier: O_RING_1

**Summary**: This series of papers is devoted to the notion of the ordered ring, and one of its most important cases: the notion of ordered field. It follows the results of \cite{SZMIELEW:1}. The idea of the notion of order in the ring is based on that of positive cone i.e. the set of positive elements. Positive cone has to contain at least squares of all elements, and be closed under sum and product. Therefore the key notions of this theory are that of square, sum of squares, product of squares, etc. and finally elements generated from squares by means of sums and products. Part I contains definitions of all those key notions and inclusions between them.

- Michal Muzalewski, Leslaw W. Szczerba.
Ordered Rings -- Part II,
Formalized Mathematics 2(2), pages 247-249, 1991. MML Identifier: O_RING_2

**Summary**: This series of papers is devoted to the notion of the ordered ring, and one of its most important cases: the notion of ordered field. It follows the results of \cite{SZMIELEW:1}. The idea of the notion of order in the ring is based on that of positive cone i.e. the set of positive elements. Positive cone has to contain at least squares of all elements, and has to be closed under sum and product. Therefore the key notions of this theory are that of square, sum of squares, product of squares, etc. and finally elements generated from squares by means of sums and products. Part II contains classification of sums of such elements.

- Michal Muzalewski, Leslaw W. Szczerba.
Ordered Rings -- Part III,
Formalized Mathematics 2(2), pages 251-253, 1991. MML Identifier: O_RING_3

**Summary**: This series of papers is devoted to the notion of the ordered ring, and one of its most important cases: the notion of ordered field. It follows the results of \cite{SZMIELEW:1}. The idea of the notion of order in the ring is based on that of positive cone i.e. the set of positive elements. Positive cone has to contain at least squares of all elements, and be closed under sum and product. Therefore the key notions of this theory are that of square, sum of squares, product of squares, etc. and finally elements generated from squares by means of sums and products. Part III contains classification of products of such elements.

- Michal Muzalewski, Wojciech Skaba.
N-Tuples and Cartesian Products for n$=$5,
Formalized Mathematics 2(2), pages 255-258, 1991. MML Identifier: MCART_2

**Summary**: This article defines ordered $n$-tuples, projections and Cartesian products for $n=5$. We prove many theorems concerning the basic properties of the $n$-tuples and Cartesian products that may be utilized in several further, more challenging applications. A few of these theorems are a strightforward consequence of the regularity axiom. The article originated as an upgrade of the article \cite{MCART_1.ABS}.

- Michal Muzalewski, Wojciech Skaba.
Ternary Fields,
Formalized Mathematics 2(2), pages 259-261, 1991. MML Identifier: ALGSTR_3

**Summary**: This article contains part 3 of the set of papers concerning the theory of algebraic structures, based on the book \cite[pp. 13--15]{SZMIELEW:1} (pages 6--8 for English edition).\par First the basic structure $\langle F, 0, 1, T\rangle$ is defined, where $T$ is a ternary operation on $F$ (three argument operations have been introduced in the article \cite{MULTOP_1.ABS}. Following it, the basic axioms of a ternary field are displayed, the mode is defined and its existence proved. The basic properties of a ternary field are also contemplated there.}

- Jozef Bialas.
The $\sigma$-additive Measure Theory,
Formalized Mathematics 2(2), pages 263-270, 1991. MML Identifier: MEASURE1

**Summary**: The article contains definition and basic properties of $\sigma$-additive, nonnegative measure, with values in $\overline{\Bbb R}$, the enlarged set of real numbers, where $\overline{\Bbb R}$ denotes set $\overline{\Bbb R} = {\Bbb R} \cup \{-\infty,+\infty\}$ - by \cite{SIKORSKI:1}. We present definitions of $\sigma$-field of sets, $\sigma$-additive measure, measurable sets, measure zero sets and the basic theorems describing relationships between the notion mentioned above. The work is the third part of the series of articles concerning the Lebesgue measure theory.

- Eugeniusz Kusak, Wojciech Leonczuk.
Incidence Projective Space (a reduction theorem in a plane),
Formalized Mathematics 2(2), pages 271-274, 1991. MML Identifier: PROJRED1

**Summary**: The article begins with basic facts concerning arbitrary projective spaces. Further we are concerned with Fano projective spaces (we prove it has rank at least four). Finally we restrict ourselves to Desarguesian planes; we define the notion of perspectivity and we prove the reduction theorem for projectivities with concurrent axes.

- Michal Muzalewski, Wojciech Skaba.
Groups, Rings, Left- and Right-Modules,
Formalized Mathematics 2(2), pages 275-278, 1991. MML Identifier: MOD_1

**Summary**: The notion of group was defined as a group structure introduced in the article \cite{VECTSP_1.ABS}. The article contains the basic properties of groups, rings, left- and right-modules of an associative ring.

- Michal Muzalewski, Wojciech Skaba.
Finite Sums of Vectors in Left Module over Associative Ring,
Formalized Mathematics 2(2), pages 279-282, 1991. MML Identifier: LMOD_1

**Summary**:

- Michal Muzalewski, Wojciech Skaba.
Submodules and Cosets of Submodules in Left Module over Associative Ring,
Formalized Mathematics 2(2), pages 283-287, 1991. MML Identifier: LMOD_2

**Summary**:

- Michal Muzalewski, Wojciech Skaba.
Operations on Submodules in Left Module over Associative Ring,
Formalized Mathematics 2(2), pages 289-293, 1991. MML Identifier: LMOD_3

**Summary**:

- Michal Muzalewski, Wojciech Skaba.
Linear Combinations in Left Module over Associative Ring,
Formalized Mathematics 2(2), pages 295-300, 1991. MML Identifier: LMOD_4

**Summary**:

- Michal Muzalewski, Wojciech Skaba.
Linear Independence in Left Module over Domain,
Formalized Mathematics 2(2), pages 301-303, 1991. MML Identifier: LMOD_5

**Summary**: Notion of a submodule generated by a set of vectors and linear independence of a set of vectors. A few theorems originated as a generalization of the theorems from the article \cite{VECTSP_7.ABS}.

- Jan Popiolek, Andrzej Trybulec.
Calculus of Propositions,
Formalized Mathematics 2(2), pages 305-307, 1991. MML Identifier: PROCAL_1

**Summary**: Continues the analysis of classical language of first order (see \cite{QC_LANG1.ABS}, \cite{QC_LANG2.ABS}, \cite{CQC_LANG.ABS}, \cite{CQC_THE1.ABS}, \cite{LUKASI_1.ABS}). Three connectives: truth, negation and conjuction are primary (see \cite{QC_LANG1.ABS}). The others (alternative, implication and equivalence) are defined with respect to them (see \cite{QC_LANG2.ABS}). We prove some important tautologies of the calculus of propositions. Most of them are given as the axioms of classical logical calculus (see \cite{GRZEG1}). In the last part of our article we give some basic rules of inference.

- Agata Darmochwal.
Calculus of Quantifiers. Deduction Theorem,
Formalized Mathematics 2(2), pages 309-312, 1991. MML Identifier: CQC_THE2

**Summary**: Some tautologies of the Classical Quantifier Calculus. The deduction theorem is also proved.

- Stanislawa Kanas, Jan Stankiewicz.
Metrics in Cartesian Product,
Formalized Mathematics 2(2), pages 193-197, 1991. MML Identifier: METRIC_3
- Number 3
- Henryk Oryszczyszyn, Krzysztof Prazmowski.
A construction of analytical Ordered Trapezium Spaces,
Formalized Mathematics 2(3), pages 315-322, 1991. MML Identifier: ANALTRAP

**Summary**:

- Henryk Oryszczyszyn, Krzysztof Prazmowski.
A construction of analytical Ordered Trapezium Spaces,
Formalized Mathematics 2(3), pages 315-322, 1991. MML Identifier: GEOMTRAP

**Summary**: We define, in a given real linear space, the midpoint operation on vectors and, with the help of the notions of directed parallelism of vectors and orthogonality of vectors, we define the relation of directed trapezium. We consider structures being enrichments of affine structures by one binary operation, together with a function which assigns to every such a structure its ``affine" reduct. Theorems concerning midpoint operation and trapezium relation are proved which enables us to introduce an abstract notion of (regular in fact) ordered trapezium space with midpoint, ordered trapezium space, and (unordered) trapezium space.

- Eugeniusz Kusak, Wojciech Leonczuk, Krzysztof Prazmowski.
On Projections in Projective Planes (Part II ),
Formalized Mathematics 2(3), pages 323-329, 1991. MML Identifier: PROJRED2

**Summary**: We study in greater details projectivities on Desarguesian projective planes. We are particularly interested in the situation when the composition of given two projectivities can be replaced by another two, with given axis or centre of one of them.

- Jolanta Swierzynska, Bogdan Swierzynski.
Metric-Affine Configurations in Metric Affine Planes -- Part I,
Formalized Mathematics 2(3), pages 331-334, 1991. MML Identifier: CONAFFM

**Summary**: We introduce several configurational axioms for metric affine planes such as theorem on three perpendiculars, orthogonalization of major Desargues Axiom, orthogonalization of the trapezium variant of Desargues Axiom, axiom on parallel projection together with its indirect forms. For convenience we also consider affine Major Desargues Axiom. The aim is to prove logical relationships which hold between the introduced statements.

- Jolanta Swierzynska, Bogdan Swierzynski.
Metric-Affine Configurations in Metric Affine Planes -- Part II,
Formalized Mathematics 2(3), pages 335-340, 1991. MML Identifier: CONMETR

**Summary**: A continuation of \cite{CONAFFM.ABS}. We introduce more configurational axioms i.e. orthogonalizations of ``scherungssatzes" (direct and indirect), ``Scherungssatz" with orthogonal axes, Pappus axiom with orthogonal axes; we also consider the affine Major Pappus Axiom and affine minor Desargues Axiom. We prove a number of implications which hold between the above axioms.

- Krzysztof Prazmowski.
Fanoian, Pappian and Desarguesian Affine Spaces,
Formalized Mathematics 2(3), pages 341-346, 1991. MML Identifier: PAPDESAF

**Summary**: We introduce basic types of affine spaces such as Desarguesian, Fanoian, Pappian, and translation affine and ordered affine sapces and we prove that suitably choosen analytically defined affine structures satisfy the required properties.

- Krzysztof Prazmowski, Krzysztof Radziszewski.
Elementary Variants of Affine Configurational Theorems,
Formalized Mathematics 2(3), pages 347-348, 1991. MML Identifier: PARDEPAP

**Summary**: We present elementary versions of Pappus, Major Desargues and Minor Desargues Axioms (i.e. statements formulated entirely in the language of points and parallelity of segments). Evidently they are consequences of appropriate configurational axioms introduced in the article \cite{AFF_2.ABS}. In particular it follows that there exists an affine plane satisfying all of them.

- Eugeniusz Kusak, Krzysztof Radziszewski.
Semi_Affine Space,
Formalized Mathematics 2(3), pages 349-356, 1991. MML Identifier: SEMI_AF1

**Summary**: A brief survey on semi-affine geometry, which results from the classical Pappian and Desarguesian affine (dimension free) geometry by weakening the so called trapezium axiom. With the help of the relation of parallelogram in every semi-affine space we define the operation of ``addition" of ``vectors". Next we investigate in greater details the relation of (affine) trapezium in such spaces.

- Wojciech Leonczuk, Henryk Oryszczyszyn, Krzysztof Prazmowski.
Planes in Affine Spaces,
Formalized Mathematics 2(3), pages 357-363, 1991. MML Identifier: AFF_4

**Summary**: We introduce the notion of plane in affine space and investigate fundamental properties of them. Further we introduce the relation of parallelism defined for arbitrary subsets. In particular we are concerned with parallelisms which hold between lines and planes and between planes. We also define a function which assigns to every line and every point the unique line passing through the point and parallel to the given line. With the help of the introduced notions we prove that every at least 3-dimensional affine space is Desarguesian and translation.

- Krzysztof Hryniewiecki.
Graphs,
Formalized Mathematics 2(3), pages 365-370, 1991. MML Identifier: GRAPH_1

**Summary**: Definitions of graphs are introduced and their basic properties are proved. The following notions related to graph theory are introduced: subgraph, finite graph, chain and oriented chain - as a finite sequence of edges, path and oriented path - as a finite sequence of different edges, cycle and oriented cycle, incidency of graph's vertices, a sum of two graphs, a degree of a vertice, a set of all subgraphs of a graph. Many ideas of this article have been taken from \cite{Wilson}.

- Andrzej Kondracki.
Mostowski's Fundamental Operations -- Part I,
Formalized Mathematics 2(3), pages 371-375, 1991. MML Identifier: ZF_FUND1

**Summary**: In the chapter II.4 of his book \cite {MOST:1} A.~Mostowski introduces what he calls fundamental operations:\\ \indent $A_{1}(a,b)=\lbrace\lbrace\langle0,x\rangle,\langle1,y\rangle\rbrace: x\in y \wedge x\in a \wedge y\in a \rbrace$,\\ \indent $A_{2}(a,b)=\lbrace a,b\rbrace$,\\ \indent $A_{3}(a,b)=\bigcup a$,\\ \indent $A_{4}(a,b)=\lbrace\lbrace\langle x,y\rangle\rbrace: x\in a \wedge y\in b \rbrace$,\\ \indent $A_{5}(a,b)=\lbrace x\cup y : x\in a \wedge y\in b \rbrace$,\\ \indent $A_{6}(a,b)=\lbrace x\setminus y : x\in a \wedge y\in b \rbrace$,\\ \indent $A_{7}(a,b)=\lbrace x\circ y : x\in a \wedge y\in b \rbrace$.\\ He proves that if a non-void class is closed under these operations then it is predicatively closed. Then he formulates sufficient criteria for a class to be a model of ZF set theory (theorem 4.12). \par The article includes the translation of this part of Mostowski's book. The fundamental operations are defined (to be precise not these operations, but the notions of closure of a class with respect to them). Some properties of classes closed under these operations are proved. At last it is proved that if a non-void class $X$ is closed with respect to the operations $A_{1}-A_{7}$ then $D_{H}(a)\in X$ for every $a$ in $X$ and every $H$ being formula of ZF language ($D_{H}(a)$ consists of all finite sequences with terms belonging to $a$ which satisfy $H$ in $a$).

- Henryk Oryszczyszyn, Krzysztof Prazmowski.
A Projective Closure and Projective Horizon of an Affine Space,
Formalized Mathematics 2(3), pages 377-384, 1991. MML Identifier: AFPROJ

**Summary**: With every affine space $A$ we correlate two incidence structures. The first, called Inc-ProjSp($A$), is the usual projective closure of $A$, i.e. the structure obtained from $A$ by adding directions of lines and planes of $A$. The second, called projective horizon of $A$, is the structure built from directions. We prove that Inc-ProjSp($A$) is always a projective space, and projective horizon of $A$ is a projective space provided $A$ is at least 3-dimensional. Some evident relationships between projective and affine configurational axioms that may hold in $A$ and in Inc-ProjSp($A$) are established.

- Stanislaw T. Czuba.
Schemes,
Formalized Mathematics 2(3), pages 385-391, 1991. MML Identifier: SCHEMS_1

**Summary**: Some basic schemes of quantifier calculus are proved.

- Andrzej Trybulec.
Algebra of Normal Forms Is a Heyting Algebra,
Formalized Mathematics 2(3), pages 393-396, 1991. MML Identifier: HEYTING1

**Summary**: We prove that the lattice of normal forms over an arbitrary set, introduced in \cite{NORMFORM.ABS}, is an implicative lattice. The relative pseudo-complement $\alpha\Rightarrow\beta$ is defined as $\bigsqcup_{\alpha_1\cup\alpha_2=\alpha}-\alpha_1\sqcap \alpha_2\rightarrowtail\beta$, where $-\alpha$ is the pseudo-complement of $\alpha$ and $\alpha\rightarrowtail\beta$ is a rather strong implication introduced in this paper.

- Grzegorz Bancerek.
K\"onig's Lemma,
Formalized Mathematics 2(3), pages 397-402, 1991. MML Identifier: TREES_2

**Summary**: A continuation of \cite{TREES_1.ABS}. The notion of finite--order trees, succesors of an element of a tree, and chains, levels and branches of a tree are introduced. That notion has been used to formalize K\"onig's Lemma which claims that there is a infinite branch of a finite-order tree if the tree has arbitrary long finite chains. Besides, the concept of decorated trees is introduced and some concepts dealing with trees are applied to decorated trees.

- Jaroslaw Kotowicz.
Monotonic and Continuous Real Function,
Formalized Mathematics 2(3), pages 403-405, 1991. MML Identifier: FCONT_3

**Summary**: A continuation of \cite{FCONT_1.ABS} and \cite{FCONT_2.ABS}. We prove a few theorems about real functions monotonic and continuous on interval, on halfline and on the set of real numbers and continuity of the inverse function. At the beginning of the paper we show some facts about topological properties of the set of real numbers, halflines and intervals which rather belong to \cite{RCOMP_1.ABS}.

- Jaroslaw Kotowicz, Konrad Raczkowski.
Real Function Differentiability -- Part II,
Formalized Mathematics 2(3), pages 407-411, 1991. MML Identifier: FDIFF_2

**Summary**: A continuation of \cite{FDIFF_1.ABS}. We prove equivalent definition of the derivative of the real function at the point and theorems about derivative of composite functions, inverse function and derivative of quotient of two functions. At the beginning of the paper a few facts which rather belong to \cite{SEQ_2.ABS}, \cite{SEQM_3.ABS} and \cite{SEQ_4.ABS} are proved.

- Wojciech Zielonka.
Preliminaries to the Lambek calculus,
Formalized Mathematics 2(3), pages 413-418, 1991. MML Identifier: PRELAMB

**Summary**: Some preliminary facts concerning completeness and decidability problems for the Lambek Calculus \cite{LAMBEK:1} are proved as well as some theses and derived rules of the calculus itself.

- Czeslaw Bylinski.
Opposite Categories and Contravariant Functors,
Formalized Mathematics 2(3), pages 419-424, 1991. MML Identifier: OPPCAT_1

**Summary**: The opposite category of a category, contravariant functors and duality functors are defined.

- Grzegorz Bancerek, Andrzej Kondracki.
Mostowski's Fundamental Operations -- Part II,
Formalized Mathematics 2(3), pages 425-427, 1991. MML Identifier: ZF_FUND2

**Summary**: The article consists of two parts. The first part is translation of chapter II.3 of \cite{MOST:1}. A section of $D_{H}(a)$ determined by $f$ (symbolically $S_{H}(a,f)$) and a notion of predicative closure of a class are defined. It is proved that if following assumptions are satisfied: (o) A=\bigcup_{\xi}A_{\xi}, (i) $A_{\xi} \subset A_{\eta}$ for $\xi < \eta$, (ii) $A_{\lambda}=\bigcup_{\xi<\lambda}A_{\lambda}$ ($\lambda$ is a limit number), (iii) $A_{\xi}\in A$, (iv) $A_{\xi}$ is transitive, (v) $(x,y\in A) \rightarrow (x\cap y\in A)$, (vi) $A$ is predicatively closed, then the axiom of power sets and the axiom of substitution are valid in $A$. The second part is continuation of \cite{ZF_FUND1.ABS}. It is proved that if a non-void, transitive class is closed with respect to the operations $A_{1}-A_{7}$ then it is predicatively closed. At last sufficient criteria for a class to be a model of ZF-theory are formulated: if $A_{\xi}$ satisfies o -- iv and $A$ is closed under the operations $A_{1}-A_{7}$ then $A$ is a model of ZF.

- Henryk Oryszczyszyn, Krzysztof Prazmowski.
Fundamental Types of Metric Affine Spaces,
Formalized Mathematics 2(3), pages 429-432, 1991. MML Identifier: EUCLMETR

**Summary**: We distinguish in the class of metric affine spaces some fundamental types of them. First we can assume the underlying affine space to satisfy classical affine configurational axiom; thus we come to Pappian, Desarguesian, Moufangian, and translation spaces. Next we distinguish the spaces satisfying theorem on three perpendiculars and the homogeneous spaces; these properties directly refer to some axioms involving orthogonality. Some known relationships between the introduced classes of structures are established. We also show that the commonly investigated models of metric affine geometry constructed in a real linear space with the help of a symmetric bilinear form belong to all the classes introduced in the paper.

- Grzegorz Bancerek.
Filters -- Part II. Quotient Lattices Modulo Filters and Direct Product of Two Lattices,
Formalized Mathematics 2(3), pages 433-438, 1991. MML Identifier: FILTER_1

**Summary**: Binary and unary operation preserving binary relations and quotients of those operations modulo equivalence relations are introduced. It is shown that the quotients inherit some important properties (commutativity, associativity, distributivity, etc.). Based on it, the quotient (also called factor) lattice modulo a filter (i.e. modulo the equivalence relation w.r.t the filter) is introduced. Similarly, some properties of the direct product of two binary (unary) operations are present and then the direct product of two lattices is introduced. Besides, the heredity of distributivity, modularity, completeness, etc., for the product of lattices is also shown. Finally, the concept of isomorphic lattices is introduced, and there is shown that every Boolean lattice $B$ is isomorphic with the direct product of the factor lattice $B/[a]$ and the lattice latt$[a]$, where $a$ is an element of $B$.

- Jolanta Swierzynska, Bogdan Swierzynski.
Shear Theorems and their role in Affine Geometry,
Formalized Mathematics 2(3), pages 439-444, 1991. MML Identifier: CONMETR1

**Summary**: Investigations on affine shear theorems, major and minor, direct and indirect. We prove logical relationships which hold between these statements and between them and other classical affine configurational axioms (eg. minor and major Pappus Axiom, Desargues Axioms et al.). For the shear, Desargues, and Pappus Axioms formulated in terms of metric affine spaces we prove they are equivalent to corresponding statements formulated in terms of affine reduct of the given space.

- Henryk Oryszczyszyn, Krzysztof Prazmowski.
A construction of analytical Ordered Trapezium Spaces,
Formalized Mathematics 2(3), pages 315-322, 1991. MML Identifier: ANALTRAP
- Number 4
- Konrad Raczkowski, Andrzej Ndzusiak.
Series,
Formalized Mathematics 2(4), pages 449-452, 1991. MML Identifier: SERIES_1

**Summary**: The article contains definitions and properties of convergent serieses.

- Marek Chmur.
The Lattice of Natural Numbers and The Sublattice of it. The Set of Prime Numbers,
Formalized Mathematics 2(4), pages 453-459, 1991. MML Identifier: NAT_LAT

**Summary**: Basic properties of the least common multiple and the greatest common divisor. The lattice of natural numbers (${\rm L}_{\Bbb N}$) and the lattice of natural numbers greater than zero (${\rm L}_{\Bbb N^+}$) are constructed. The notion of the sublattice of the lattice of natural numbers is given. Some facts about it are proved. The last part of the article deals with some properties of prime numbers and with the notions of the set of prime numbers and the $n$-th prime number. It is proved that the set of prime numbers is infinite.

- Wojciech A. Trybulec.
Commutator and Center of a Group,
Formalized Mathematics 2(4), pages 461-466, 1991. MML Identifier: GROUP_5

**Summary**: We introduce the notions of commutators of element, subgroups of a group, commutator of a group and center of a group. We prove P.Hall identity. The article is based on \cite{KARGAP:1}.

- Andrzej Trybulec.
Natural Transformations. Discrete Categories,
Formalized Mathematics 2(4), pages 467-474, 1991. MML Identifier: NATTRA_1

**Summary**: We present well known concepts of category theory: natural transformations and functor categories, and prove propositions related to. Because of the formalization it proved to be convenient to introduce some auxiliary notions, for instance: transformations. We mean by a transformation of a functor $F$ to a functor $G$, both covariant functors from $A$ to $B$, a function mapping the objects of $A$ to the morphisms of $B$ and assigning to an object $a$ of $A$ an element of $\mathop{\rm Hom}(F(a),G(a))$. The material included roughly corresponds to that presented on pages 18,129--130,137--138 of the monography (\cite{SEMAD}). We also introduce discrete categories and prove some propositions to illustrate the concepts introduced.

- Katarzyna Jankowska.
Matrices. Abelian Group of Matrices,
Formalized Mathematics 2(4), pages 475-480, 1991. MML Identifier: MATRIX_1

**Summary**: The basic conceptions of matrix algebra are introduced. The matrix is introduced as the finite sequence of sequences with the same length, i.e. as a sequence of lines. There are considered matrices over a field, and the fact that these matrices with addition form an Abelian group is proved.

- Leszek Borys.
Paracompact and Metrizable Spaces,
Formalized Mathematics 2(4), pages 481-485, 1991. MML Identifier: PCOMPS_1

**Summary**: We give an example of a compact space. Next we define a locally finite subset family of a topological space and a paracompact topological space. An open sets family of a metric space we define next and it has been shown that the metric space with any open sets family is a topological space. Next we define metrizable space.

- Michal Muzalewski.
Atlas of Midpoint Algebra,
Formalized Mathematics 2(4), pages 487-491, 1991. MML Identifier: MIDSP_2

**Summary**: This article is a continuation of \cite{MIDSP_1.ABS}. We have established a one-to-one correspondence between midpoint algebras and groups with the operator 1/2. In general we shall say that a given midpoint algebra $M$ and a group $V$ are $w$-assotiated iff $w$ is an atlas from $M$ to $V$. At the beginning of the paper a few facts which rather belong to \cite{VECTSP_1.ABS}, \cite{MOD_1.MIZ} are proved.

- Jozef Bialas.
Several Properties of the $\sigma$-additive Measure,
Formalized Mathematics 2(4), pages 493-497, 1991. MML Identifier: MEASURE2

**Summary**: A continuation of \cite{MEASURE1.ABS}. The paper contains the definition and basic properties of a $\sigma$-additive, nonnegative measure, with values in $\overline{\Bbb R}$, the enlarged set of real numbers, where $\overline{\Bbb R}$ denotes set $\overline{\Bbb R} = {\Bbb R} \cup \{-\infty,+\infty\}$ --- by R.~Sikorski \cite{SIKORSKI:1}. Some simple theorems concerning basic properties of a $\sigma$-additive measure, measurable sets, measure zero sets are proved. The work is the fourth part of the series of articles concerning the Lebesgue measure theory.

- Stanislawa Kanas, Adam Lecko.
Metrics in the Cartesian Product -- Part II,
Formalized Mathematics 2(4), pages 499-504, 1991. MML Identifier: METRIC_4

**Summary**: A continuation of \cite{METRIC_3.ABS}. It deals with the method of creation of the distance in the Cartesian product of metric spaces. The distance between two points belonging to Cartesian product of metric spaces has been defined as square root of the sum of squares of distances of appropriate coordinates (or projections) of these points. It is shown that product of metric spaces with such a distance is a metric space. Examples of metric spaces defined in this way are given.

- Alicia de la Cruz.
Fix Point Theorem for Compact Spaces,
Formalized Mathematics 2(4), pages 505-506, 1991. MML Identifier: ALI2

**Summary**: The Banach theorem in compact metric spaces is proved.

- Jan Popiolek.
Quadratic Inequalities,
Formalized Mathematics 2(4), pages 507-509, 1991. MML Identifier: QUIN_1

**Summary**: Consider a quadratic trinomial of the form $P(x)=ax^2+bx+c$, where $a\ne 0$. The determinant of the equation $P(x)=0$ is of the form $\Delta(a,b,c)=b^2-4ac$. We prove several quadratic inequalities when $\Delta(a,b,c)<0$, $\Delta(a,b,c)=0$ and $\Delta(a,b,c)>0$.

- Jan Popiolek.
Introduction to Banach and Hilbert Spaces -- Part I,
Formalized Mathematics 2(4), pages 511-516, 1991. MML Identifier: BHSP_1

**Summary**: Basing on the notion of real linear space (see \cite{RLVECT_1.ABS}) we introduce real unitary space. At first, we define the scalar product of two vectors and examine some of its properties. On the base of this notion we introduce the norm and the distance in real unitary space and study properties of these concepts. Next, proceeding from the definition of the sequence in real unitary space and basic operations on sequences we prove several theorems which will be used in our further considerations.

- Jan Popiolek.
Introduction to Banach and Hilbert Spaces -- Part II,
Formalized Mathematics 2(4), pages 517-521, 1991. MML Identifier: BHSP_2

**Summary**: A continuation of \cite{BHSP_1.ABS}. It contains the definitions of the convergent sequence and limit of the sequence. The convergence with respect to the norm and the distance is also introduced. Last part of this article is devoted to the following concepts: ball, closed ball and sphere.

- Jan Popiolek.
Introduction to Banach and Hilbert Spaces -- Part III,
Formalized Mathematics 2(4), pages 523-526, 1991. MML Identifier: BHSP_3

**Summary**: The article is a continuation of \cite{BHSP_1.ABS} and of \cite{BHSP_2.ABS}. First we define the following concepts: the Cauchy sequence, the bounded sequence and the subsequence. The last part of this article contains definitions of the complete space and the Hilbert space.

- Czeslaw Bylinski.
Category Ens,
Formalized Mathematics 2(4), pages 527-533, 1991. MML Identifier: ENS_1

**Summary**: If $V$ is any non-empty set of sets, we define $\hbox{\bf Ens}_V$ to be the category with the objects of all sets $X \in V$, morphisms of all mappings from $X$ into $Y$, with the usual composition of mappings. By a mapping we mean a triple $\langle X,Y,f \rangle$ where $f$ is a function from $X$ into $Y$. The notations and concepts included corresponds to that presented in \cite{SEMAD}, \cite{MacLane:1}. We also introduce representable functors to illustrate properties of the category {\bf Ens}.

- Andrzej Trybulec.
A Borsuk Theorem on Homotopy Types,
Formalized Mathematics 2(4), pages 535-545, 1991. MML Identifier: BORSUK_1

**Summary**: We present a Borsuk's theorem published first in \cite{BORSUK:3} (compare also \cite[pages 119--120]{BORSUK:2}). It is slightly generalized, the assumption of the metrizability is omitted. We introduce concepts needed for the formulation and the proofs of the theorems on upper semi-continuous decompositions, retracts, strong deformation retract. However, only those facts that are necessary in the proof have been proved.

- Grzegorz Bancerek.
Cartesian Product of Functions,
Formalized Mathematics 2(4), pages 547-552, 1991. MML Identifier: FUNCT_6

**Summary**: A supplement of \cite{CARD_3.ABS} and \cite{FUNCT_5.ABS}, i.e. some useful and explanatory properties of the product and also the curried and uncurried functions are shown. Besides, the functions yielding functions are considered: two different products and other operation of such functions are introduced. Finally, two facts are presented: quasi-distributivity of the power of the set to other one w.r.t. the union ($X^{\biguplus_{x}f(x)} \approx \prod_{x}X^{f(x)}$) and quasi-distributivity of the product w.r.t. the raising to the power ($\prod_{x}{f(x)^X} \approx (\prod_{x}f(x))^X$).

- Alicia de la Cruz.
Introduction to Modal Propositional Logic,
Formalized Mathematics 2(4), pages 553-558, 1991. MML Identifier: MODAL_1

**Summary**:

- Alicia de la Cruz.
Totally Bounded Metric Spaces,
Formalized Mathematics 2(4), pages 559-562, 1991. MML Identifier: TBSP_1

**Summary**:

- Michal Muzalewski.
Categories of Groups,
Formalized Mathematics 2(4), pages 563-571, 1991. MML Identifier: GRCAT_1

**Summary**: We define the category of groups and its subcategories: category of Abelian groups and category of groups with the operator of $1/2$. The carriers of the groups are included in a universum. The universum is a parameter of the categories.

- Wojciech A. Trybulec, Michal J. Trybulec.
Homomorphisms and Isomorphisms of Groups. Quotient Group,
Formalized Mathematics 2(4), pages 573-578, 1991. MML Identifier: GROUP_6

**Summary**: Quotient group, homomorphisms and isomorphisms of groups are introduced. The so called isomorphism theorems are proved following \cite{KARGAP:1}.

- Michal Muzalewski.
Rings and Modules -- Part II,
Formalized Mathematics 2(4), pages 579-585, 1991. MML Identifier: MOD_2

**Summary**: We define the trivial left module, morphism of left modules and the field $Z_3$. We prove some elementary facts.

- Michal Muzalewski.
Free Modules,
Formalized Mathematics 2(4), pages 587-589, 1991. MML Identifier: MOD_3

**Summary**: We define free modules and prove that every left module over Skew-Field is free.

- Jaroslaw Zajkowski.
Oriented Metric-Affine Plane -- Part I,
Formalized Mathematics 2(4), pages 591-597, 1991. MML Identifier: ANALORT

**Summary**: We present (in Euclidean and Minkowskian geometry) definitions and some properties of oriented orthogonality relation. Next we consider consistence Euclidean space and consistence Minkowskian space.

- Agata Darmochwal.
The Euclidean Space,
Formalized Mathematics 2(4), pages 599-603, 1991. MML Identifier: EUCLID

**Summary**: The general definition of Euclidean Space.

- Agata Darmochwal, Yatsuka Nakamura.
Metric Spaces as Topological Spaces -- Fundamental Concepts,
Formalized Mathematics 2(4), pages 605-608, 1991. MML Identifier: TOPMETR

**Summary**: Some notions connected with metric spaces and the relationship between metric spaces and topological spaces. Compactness of topological spaces is transferred for the case of metric spaces \cite{KELL55}. Some basic theorems about translations of topological notions for metric spaces are proved. One-dimensional topological space ${\Bbb R^1}$ is introduced, too.

- Agata Darmochwal, Yatsuka Nakamura.
Heine--Borel's Covering Theorem,
Formalized Mathematics 2(4), pages 609-610, 1991. MML Identifier: HEINE

**Summary**: Heine--Borel's covering theorem, also known as Borel--Lebesgue theorem (\cite{BOURBAKI}), is proved. Some useful theorems about real inequalities, intervals, sequences and notion of power sequence which are necessary for the theorem are also proved.

- Yatsuka Nakamura, Agata Darmochwal.
Some Facts about Union of Two Functions and Continuity of Union of Functions,
Formalized Mathematics 2(4), pages 611-613, 1991. MML Identifier: TOPMETR2

**Summary**: Proofs of two theorems connected with union of any two functions and the proofs of two theorems about continuity of the union of two continuous functions between topogical spaces. The theorem stating that union of two subsets of $R^2$, which are homeomorphic to unit interval and have only one terminal joined point is also homeomorphic to unit interval is proved, too.

- Konrad Raczkowski, Andrzej Ndzusiak.
Series,
Formalized Mathematics 2(4), pages 449-452, 1991. MML Identifier: SERIES_1
- Number 5
- Agata Darmochwal, Yatsuka Nakamura.
The Topological Space $\calE^2_\rmT$. Arcs, Line Segments and Special Polygonal Arcs,
Formalized Mathematics 2(5), pages 617-621, 1991. MML Identifier: TOPREAL1

**Summary**: The notions of arc and line segment are introduced in two-dimensional topological real space ${\cal E}^2_{\rm T}$. Some basic theorems for these notions are proved. Using line segments, the notion of special polygonal arc is defined. It has been shown that any special polygonal arc is homeomorphic to unit interval ${\Bbb I}$. The notion of unit square $\square_{\cal E^{2}_{\rm T}}$ has been also introduced and some facts about it have been proved.

- Dariusz Surowik.
Cyclic Groups and Some of Their Properties -- Part I,
Formalized Mathematics 2(5), pages 623-627, 1991. MML Identifier: GR_CY_1

**Summary**: Some properties of finite groups are proved. The notion of cyclic group is defined next, some cyclic groups are given, for example the group of integers with addition operations. Chosen properties of cyclic groups are proved next.

- Andrzej Trybulec.
Isomorphisms of Categories,
Formalized Mathematics 2(5), pages 629-634, 1991. MML Identifier: ISOCAT_1

**Summary**: We continue the development of the category theory basically following \cite{SEMAD} (compare also \cite{MacLane:1}). We define the concept of isomorphic categories and prove basic facts related, e.g. that the Cartesian product of categories is associative up to the isomorphism. We introduce the composition of a functor and a transformation, and of transformation and a functor, and afterwards we define again those concepts for natural transformations. Let us observe, that we have to duplicate those concepts because of the permissiveness: if a functor $F$ is not naturally transformable to $G$, then natural transformation from $F$ to $G$ has no fixed meaning, hence we cannot claim that the composition of it with a functor as a transformation results in a natural transformation. We define also the so called horizontal composition of transformations (\cite{SEMAD}, p.~140, exercise {\bf 4}.{\bf 2},{\bf 5}(C)) and prove {\em interchange law} (\cite{MacLane:1}, p.44). We conclude with the definition of equivalent categories.

- Agata Darmochwal, Andrzej Trybulec.
Similarity of Formulae,
Formalized Mathematics 2(5), pages 635-642, 1991. MML Identifier: CQC_SIM1

**Summary**: The main objective of the paper is to define the concept of the similarity of formulas. We mean by similar formulas the two formulas that differs only in the names of bound variables. Some authors (compare \cite{RASIOWA-SIKOR}) call such formulas {\em congruent}. We use the word {\em similar} following \cite{POGO:1}, \cite{POGO:2}, \cite{POGORZELSKI.1975}. The concept is unjustfully neglected in many logical handbooks. It is intuitively quite clear, however the exact definition is not simple. As far as we know, only W.A.~Pogorzelski and T.~Prucnal \cite{POGORZELSKI.1975} define it in the precise way. We follow basically the Pogorzelski's definition (compare \cite{POGO:1}). We define renumeration of bound variables and we say that two formulas are similar if after renumeration are equal. Therefore we need a rule of chosing bound variables independent of the original choice. Quite obvious solution is to use consecutively variables $x_{k+1},x_{k+2},\dots$, where $k$ is the maximal index of free variable occurring in the formula. Therefore after the renumeration we get the new formula in which different quantifiers bind different variables. It is the reason that the result of renumeration applied to a formula $\varphi$ we call {\em $\varphi$ with variables separated}.

- Michal Muzalewski.
Category of Rings,
Formalized Mathematics 2(5), pages 643-648, 1991. MML Identifier: RINGCAT1

**Summary**: We define the category of non-associative rings. The carriers of the rings are included in a universum. The universum is a parameter of the category.

- Michal Muzalewski.
Category of Left Modules,
Formalized Mathematics 2(5), pages 649-652, 1991. MML Identifier: MODCAT_1

**Summary**: We define the category of left modules over an associative ring. The carriers of the modules are included in a universum. The universum is a parameter of the category.

- Ewa Burakowska, Beata Madras.
Real Function One-Side Differentiability,
Formalized Mathematics 2(5), pages 653-656, 1991. MML Identifier: FDIFF_3

**Summary**: We define real function one-side differentiability and one-side continuity. Main properties of one-side differentiability function are proved. Connections between one-side differential and differential real function at the point are demonstrated.

- Stanislawa Kanas, Adam Lecko.
Sequences in Metric Spaces,
Formalized Mathematics 2(5), pages 657-661, 1991. MML Identifier: METRIC_6

**Summary**: Sequences in metric spaces are defined. The article contains definitions of bounded, convergent, Cauchy sequences. The subsequences are introduced too. Some theorems concerning sequences are proved.

- Agata Darmochwal, Yatsuka Nakamura.
The Topological Space $\calE^2_\rmT$. Simple Closed Curves,
Formalized Mathematics 2(5), pages 663-664, 1991. MML Identifier: TOPREAL2

**Summary**: Continuation of \cite{TOPREAL1.ABS}. The fact that the unit square is compact is shown in the beginning of the article. Next the notion of simple closed curve is introduced. It is proved that any simple closed curve can be divided into two independent parts which are homeomorphic to unit interval ${\Bbb I}$.

- Zbigniew Karno.
Separated and Weakly Separated Subspaces of Topological Spaces,
Formalized Mathematics 2(5), pages 665-674, 1991. MML Identifier: TSEP_1

**Summary**: A new concept of weakly separated subsets and subspaces of topological spaces is described in Mizar formalizm. Based on \cite{KURAT:2}, in comparison with the notion of separated subsets (subspaces), some properties of such subsets (subspaces) are discussed. Some necessary facts concerning closed subspaces, open subspaces and the union and the meet of two subspaces are also introduced. To present the main theorems we first formulate basic definitions. Let $X$ be a topological space. Two subsets $A_1$ and $A_2$ of $X$ are called {\em weakly separated} if $A_1 \setminus A_2$ and $A_2 \setminus A_1$ are separated. Two subspaces $X_{1}$ and $X_{2}$ of $X$ are called {\em weakly separated} if their carriers are weakly separated. The following theorem contains a useful characterization of weakly separated subsets in the special case when $A_{1} \cup A_{2}$ is equal to the carrier of $X$. {\em $A_{1}$ and $A_{2}$ are weakly separated iff there are such subsets of $X$, $C_{1}$ and $C_{2}$ closed (open) and $C$ open (closed, respectively), that $A_{1} \cup A_{2} = C_{1} \cup C_{2} \cup C$, $C_{1} \subset A_{1}$, $C_{2} \subset A_{2}$ and $C \subset A_{1} \cap A_{2}$}. Next theorem divided into two parts contains similar characterization of weakly separated subspaces in the special case when the union of $X_1$ and $X_2$ is equal to $X$. {\em If $X_{1}$ meets $X_{2}$, then $X_1$ and $X_2$ are weakly separated iff either $X_{1}$ is a subspace of $X_2$ or $X_2$ is a subspace of $X_{1}$ or there are such open (closed) subspaces $Y_1$ and $Y_2$ of $X$ that $Y_1$ is a subspace of $X_1$ and $Y_2$ is a subspace of $X_2$ and either $X$ is equal to the union of $Y_1$ and $Y_2$ or there is a(n) closed (open, respectively) subspace $Y$ of $X$ being a subspace of the meet of $X_1$ and $X_2$ and with the property that $X$ is the union of all $Y_1$, $Y_2$ and $Y$}. {\em If $X_1$ misses $X_{2}$, then $X_1$ and $X_2$ are weakly separated iff $X_1$ and $X_2$ are open (closed) subspaces of $X$}. Moreover, the following simple characterization of separated subspaces by means of weakly separated ones is obtained. {\em $X_1$ and $X_2$ are separated iff there are weakly separated subspaces $Y_1$ and $Y_2$ of $X$ such that $X_1$ is a subspace of $Y_1$, $X_2$ is a subspace of $Y_2$ and either $Y_1$ misses $Y_2$ or the meet of $Y_1$ and $Y_2$ misses the union of $X_1$ and $X_2$}.

- Malgorzata Korolkiewicz.
The de l'Hospital Theorem,
Formalized Mathematics 2(5), pages 675-678, 1991. MML Identifier: L_HOSPIT

**Summary**: List of theorems concerning the de l'Hospital Theorem. We discuss the case when both functions have the zero value at a point and when the quotient of their differentials is convergent at this point.

- Grzegorz Bancerek, Agata Darmochwal.
Comma Category,
Formalized Mathematics 2(5), pages 679-681, 1991. MML Identifier: COMMACAT

**Summary**: Comma category of two functors is introduced.

- Patricia L. Carlson, Grzegorz Bancerek.
Context-Free Grammar -- Part 1,
Formalized Mathematics 2(5), pages 683-687, 1991. MML Identifier: LANG1

**Summary**: The concept of context-free grammar and of derivability in grammar are introduced. Moreover, the language (set of finite sequences of symbols) generated by grammar and some grammars are defined. The notion convenient to prove facts on language generated by grammar with exchange of symbols on grammar of union and concatenation of languages is included.

- Jozef Bialas.
Completeness of the $\sigma$-Additive Measure. Measure Theory,
Formalized Mathematics 2(5), pages 689-693, 1991. MML Identifier: MEASURE3

**Summary**: Definitions and basic properties of a $\sigma$-additive, non-negative measure, with values in $\overline{\Bbb R}$, the enlarged set of real numbers, where $\overline{\Bbb R}$ denotes set $\overline{\Bbb R} = {\Bbb R}\cup\{-\infty,+\infty\}$ - by \cite{SIKORSKI:1}. The article includes the text being a continuation of the paper \cite{MEASURE2.ABS}. Some theorems concerning basic properties of a $\sigma$-additive measure and completeness of the measure are proved.

- Elzbieta Kraszewska, Jan Popiolek.
Series in Banach and Hilbert Spaces,
Formalized Mathematics 2(5), pages 695-699, 1991. MML Identifier: BHSP_4

**Summary**: In \cite{SERIES_1.ABS} the series of real numbers were investigated. The introduction to Banach and Hilbert spaces (\cite{BHSP_1.ABS}, \cite{BHSP_2.ABS}, \cite{BHSP_3.ABS}), enables us to arrive at the concept of series in Hilbert space. We start with the notions: partial sums of series, sum and $n$-th sum of series, convergent series (summable series), absolutely convergent series. We prove some basic theorems: the necessary condition for a series to converge, Weierstrass' test, d'Alembert's test, Cauchy's test.

- Czeslaw Bylinski.
Products and Coproducts in Categories,
Formalized Mathematics 2(5), pages 701-709, 1991. MML Identifier: CAT_3

**Summary**: A product and coproduct in categories are introduced. The concepts included correspond to that presented in \cite{SEMAD}.

- Katarzyna Jankowska.
Transpose Matrices and Groups of Permutations,
Formalized Mathematics 2(5), pages 711-717, 1991. MML Identifier: MATRIX_2

**Summary**: Some facts concerning matrices with dimension $2\times 2$ are shown. Upper and lower triangular matrices, and operation of deleting rows and columns in a matrix are introduced. Besides, we deal with sets of permutations and the fact that all permutations of finite set constitute a finite group is proved. Some proofs are based on \cite{HUNGERFORD} and \cite{LANG}.

- Grzegorz Bancerek.
Complete Lattices,
Formalized Mathematics 2(5), pages 719-725, 1991. MML Identifier: LATTICE3

**Summary**: In the first section the lattice of subsets of distinct set is introduced. The join and meet operations are, respectively, union and intersection of sets, and the ordering relation is inclusion. It is shown that this lattice is Boolean, i.e. distributive and complementary. The second section introduces the poset generated in a distinct lattice by its ordering relation. Besides, it is proved that posets which have l.u.b.'s and g.l.b.'s for every two elements generate lattices with the same ordering relations. In the last section the concept of complete lattice is introduced and discussed. Finally, the fact that the function $f$ from subsets of distinct set yielding elements of this set is a infinite union of some complete lattice, if $f$ yields an element $a$ for singleton $\{a\}$ and $f(f^\circ X) = f(\bigsqcup X)$ for every subset $X$, is proved. Some concepts and proofs are based on \cite{RASIOWA-SIKOR} and \cite{TRACZYK}.

- Agata Darmochwal, Yatsuka Nakamura.
The Topological Space $\calE^2_\rmT$. Arcs, Line Segments and Special Polygonal Arcs,
Formalized Mathematics 2(5), pages 617-621, 1991. MML Identifier: TOPREAL1
**Volume 3, 1992**- Number 1
- Zbigniew Karno.
Continuity of Mappings over the Union of Subspaces,
Formalized Mathematics 3(1), pages 1-16, 1992. MML Identifier: TMAP_1

**Summary**: Let $X$ and $Y$ be topological spaces and let $X_{1}$ and $X_{2}$ be subspaces of $X$. Let $f : X_{1} \cup X_{2} \rightarrow Y$ be a mapping defined on the union of $X_{1}$ and $X_{2}$ such that the restriction mappings $f_{\mid X_{1}}$ and $f_{\mid X_{2}}$ are continuous. It is well known that if $X_{1}$ and $X_{2}$ are both open (closed) subspaces of $X$, then $f$ is continuous (see e.g. \cite[p.106]{KURAT:2}). \par The aim is to show, using Mizar System, the following theorem (see Section 5): {\em If $X_{1}$ and $X_{2}$ are weakly separated, then $f$ is continuous} (compare also \cite[p.358]{CECH:1} for related results). This theorem generalizes the preceding one because if $X_{1}$ and $X_{2}$ are both open (closed), then these subspaces are weakly separated (see \cite{TSEP_1.ABS}). However, the following problem remains open. \begin{itemize} \item[ ] {\bf Problem 1.} Characterize the class of pairs of subspaces $X_{1}$ and $X_{2}$ of a topological space $X$ such that ($\ast$) for any topological space $Y$ and for any mapping $f : X_{1} \cup X_{2} \rightarrow Y$, $f$ is continuous if the restrictions $f_{\mid X_{1}}$ and $f_{\mid X_{2}}$ are continuous. \end{itemize} In some special case we have the following characterization: {\em $X_{1}$ and $X_{2}$ are separated iff $X_{1}$ misses $X_{2}$ and the condition} ($\ast$) {\em is fulfilled.} In connection with this fact we hope that the following specification of the preceding problem has an affirmative answer. \begin{itemize} \item[ ] {\bf Problem 2.} Suppose the condition ($\ast$) is fulfilled. Must $X_{1}$ and $X_{2}$ be weakly separated ? \end{itemize} Note that in the last section the concept of the union of two mappings is introduced and studied. In particular, all results presented above are reformulated using this notion. In the remaining sections we introduce concepts needed for the formulation and the proof of theorems on properties of continuous mappings, restriction mappings and modifications of the topology.

- Beata Perkowska.
Functional Sequence from a Domain to a Domain,
Formalized Mathematics 3(1), pages 17-21, 1992. MML Identifier: SEQFUNC

**Summary**: Definitions of functional sequences and basic operations on functional sequences from a domain to a domain, point and uniform convergence, limit of functional sequence from a domain to the set of real numbers and facts about properties of the limit of functional sequences are proved.

- Michal Muzalewski.
Reper Algebras,
Formalized Mathematics 3(1), pages 23-28, 1992. MML Identifier: MIDSP_3

**Summary**: We shall describe $n$-dimensional spaces with the reper operation \cite[pages 72--79]{MUZALEWSKI:1}. An inspiration to such approach comes from the monograph \cite{SZMIELEW:1} and so-called Leibniz program. Let us recall that the Leibniz program is a program of algebraization of geometry using purely geometric notions. Leibniz formulated his program in opposition to algebraization method developed by Descartes. The Euclidean geometry in Szmielew's approach \cite {SZMIELEW:1} is a theory of structures $\langle S$; $\parallel, \oplus, O \rangle$, where $\langle S$; $\parallel, \oplus, O \rangle$ is Desarguesian midpoint plane and $O \subseteq S\times S\times S$ is the relation of equi-orthogonal basis. Points $o, p, q$ are in relation $O$ if they form an isosceles triangle with the right angle in vertex $a$. If we fix vertices $a, p$, then there exist exactly two points $q, q'$ such that $O(apq)$, $O(apq')$. Moreover $q \oplus q' = a$. In accordance with the Leibniz program we replace the relation of equi-orthogonal basis by a binary operation $\ast : S\times S \rightarrow S$, called the reper operation. A standard model for the Euclidean geometry in the above sense is the oriented plane over the field of real numbers with the reper operations $\ast$ defined by the condition: $a \ast b = q$ iff the point $q$ is the result of rotating of $p$ about right angle around the center $a$.

- Dariusz Surowik.
Isomorphisms of Cyclic Groups. Some Properties of Cyclic Groups,
Formalized Mathematics 3(1), pages 29-32, 1992. MML Identifier: GR_CY_2

**Summary**: Some theorems and properties of cyclic groups have been proved with special regard to isomorphisms of these groups. Among other things it has been proved that an arbitrary cyclic group is isomorphic with groups of integers with addition or group of integers with addition modulo $m$. Moreover, it has been proved that two arbitrary cyclic groups of the same order are isomorphic and that the class of cyclic groups is closed in consideration of homomorphism images. Some other properties of groups of this type have been proved too.

- Andrzej Trybulec.
Some Isomorphisms Between Functor Categories,
Formalized Mathematics 3(1), pages 33-40, 1992. MML Identifier: ISOCAT_2

**Summary**: We define some well known isomorphisms between functor categories: between $A^{\mathop{\dot\circlearrowright}(o,m)}$ and $A$, between $C^{\mizleftcart A,B\mizrightcart}$ and ${(C^B)}^A$, and between ${\mizleftcart B,C\mizrightcart}^A$ and $\mizleftcart B^A,C^A\mizrightcart$. Compare \cite{SEMAD} and \cite{MacLane:1}. Unfortunately in this paper "functor" is used in two different meanings, as a lingual function and as a functor between categories.

- Toshihiko Watanabe.
The Lattice of Domains of a Topological Space,
Formalized Mathematics 3(1), pages 41-46, 1992. MML Identifier: TDLAT_1

**Summary**: Let $T$ be a topological space and let $A$ be a subset of $T$. Recall that $A$ is said to be a {\em closed domain} of $T$ if $A = \overline{{\rm Int}\,A}$ and $A$ is said to be an {\em open domain} of $T$ if $A = {\rm Int}\,\overline{A}$ (see e.g. \cite{KURAT:2}, \cite{TOPS_1.ABS}). Some simple generalization of these notions is the following one. $A$ is said to be a {\em domain} of $T$ provided ${\rm Int}\,\overline{A} \subseteq A \subseteq \overline{{\rm Int}\,A}$ (see \cite{TOPS_1.ABS} and compare \cite{ISOMICHI}). In this paper certain connections between these concepts are introduced and studied. \par Our main results are concerned with the following well--known theorems (see e.g. \cite{MOST-KURAT:3}, \cite{BIRKHOFF:1}). For a given topological space all its closed domains form a Boolean lattice, and similarly all its open domains form a Boolean lattice, too. It is proved that {\em all domains of a given topological space form a complemented lattice.} Moreover, it is shown that both {\em the lattice of open domains and the lattice of closed domains are sublattices of the lattice of all domains.} In the beginning some useful theorems about subsets of topological spaces are proved and certain properties of domains, closed domains and open domains are discussed.

- Michal Muzalewski.
Submodules,
Formalized Mathematics 3(1), pages 47-51, 1992. MML Identifier: LMOD_6

**Summary**: This article contains the notions of trivial and non-trivial leftmodules and rings, cyclic submodules and inclusion of submodules. A few basic theorems related to these notions are proved.

- Jaroslaw Zajkowski.
Oriented Metric-Affine Plane -- Part II,
Formalized Mathematics 3(1), pages 53-56, 1992. MML Identifier: DIRORT

**Summary**: Axiomatic description of properties of the oriented orthogonality relation. Next we construct (with the help of the oriented orthogonality relation) vector space and give the definitions of left-, right-, and semi-transitives.

- Michal Muzalewski.
Opposite Rings, Modules and their Morphisms,
Formalized Mathematics 3(1), pages 57-65, 1992. MML Identifier: MOD_4

**Summary**: Let $\Bbb K = \langle S; K, 0, 1, +, \cdot \rangle$ be a ring. The structure ${}^{\rm op}\Bbb K = \langle S; K, 0, 1, +, \bullet \rangle$ is called anti-ring, if $\alpha \bullet \beta = \beta \cdot \alpha$ for elements $\alpha$, $\beta$ of $K$ \cite[pages 5--7]{MUZALEWSKI:1}. It is easily seen that ${}^{\rm op}\Bbb K$ is also a ring. If $V$ is a left module over $\Bbb K$, then $V$ is a right module over ${}^{\rm op}\Bbb K$. If $W$ is a right module over $\Bbb K$, then $W$ is a left module over ${}^{\rm op}\Bbb K$. Let $K, L$ be rings. A morphism $J: K \longrightarrow L$ is called anti-homomorphism, if $J(\alpha\cdot\beta) = J(\beta)\cdot J(\alpha)$ for elements $\alpha$, $\beta$ of $K$. If $J:K \longrightarrow L$ is a homomorphism, then $J:K \longrightarrow {}^{\rm op}L$ is an anti-homomorphism. Let $K, L$ be rings, $V, W$ left modules over $K, L$ respectively and $J:K \longrightarrow L$ an anti-monomorphism. A map $f:V \longrightarrow W$ is called $J$ - semilinear, if $f(x+y) = f(x)+f(y)$ and $f(\alpha\cdot x) = J(\alpha)\cdot f(x)$ for vectors $x, y$ of $V$ and a scalar $\alpha$ of $K$.

- Jozef Bialas.
Properties of Caratheodor's Measure,
Formalized Mathematics 3(1), pages 67-70, 1992. MML Identifier: MEASURE4

**Summary**: The paper contains definitions and basic properties of Ca\-ra\-the\-o\-dory measure, with values in $\overline{\Bbb R}$, the enlarged set of real numbers, where $\overline{\Bbb R}$ denotes set $\overline{\Bbb R} = {\Bbb R}\cup\{-\infty,+\infty\}$ - by \cite{SIKORSKI:1}. The article includes the text being a continuation of the paper \cite{MEASURE3.ABS}. Caratheodory theorem and some theorems concerning basic properties of Caratheodory measure are proved. The work is the sixth part of the series of articles concerning the Lebesgue measure theory.

- Zbigniew Karno, Toshihiko Watanabe.
Completeness of the Lattices of Domains of a Topological Space,
Formalized Mathematics 3(1), pages 71-79, 1992. MML Identifier: TDLAT_2

**Summary**: Let $T$ be a topological space and let $A$ be a subset of $T$. Recall that $A$ is said to be a {\em domain} in $T$ provided ${\rm Int}\,\overline{A} \subseteq A \subseteq \overline{{\rm Int}\,A}$ (see \cite{TOPS_1.ABS} and comp. \cite{ISOMICHI}). This notion is a simple generalization of the notions of open and closed domains in $T$ (see \cite{TOPS_1.ABS}). Our main result is concerned with an extension of the following well--known theorem (see e.g. \cite{BIRKHOFF:1}, \cite{MOST-KURAT:3}, \cite{ENGEL:1}). For a given topological space the Boolean lattices of all its closed domains and all its open domains are complete. It is proved here, using Mizar System, that {\em the complemented lattice of all domains of a given topological space is complete}, too (comp. \cite{TDLAT_1.ABS}).\par It is known that both the lattice of open domains and the lattice of closed domains are sublattices of the lattice of all domains \cite{TDLAT_1.ABS}. However, the following two problems remain open. \begin{itemize} \item[ ] {\bf Problem 1.} Let $L$ be a sublattice of the lattice of all domains. Suppose $L$ is complete, is smallest with respect to inclusion, and contains as sublattices the lattice of all closed domains and the lattice of all open domains. Must $L$ be equal to the lattice of all domains~? \end{itemize} A domain in $T$ is said to be a {\em Borel domain} provided it is a Borel set. Of course every open (closed) domain is a Borel domain. It can be proved that all Borel domains form a sublattice of the lattice of domains. \begin{itemize} \item[ ] {\bf Problem 2.} Let $L$ be a sublattice of the lattice of all domains. Suppose $L$ is smallest with respect to inclusion and contains as sublattices the lattice of all closed domains and the lattice of all open domains. Must $L$ be equal to the lattice of all Borel domains~? \end{itemize} Note that in the beginning the closure and the interior operations for families of subsets of topological spaces are introduced and their important properties are presented (comp. \cite{KURAT:2}, \cite{KURAT:4}, \cite{MOST-KURAT:3}). Using these notions, certain properties of domains, closed domains and open domains are studied (comp. \cite{KURAT:4}, \cite{ENGEL:1}).

- Leszek Borys.
On Paracompactness of Metrizable Spaces,
Formalized Mathematics 3(1), pages 81-84, 1992. MML Identifier: PCOMPS_2

**Summary**: The aim is to prove, using Mizar System, one of the most important result in general topology, namely the Stone Theorem on paracompactness of metrizable spaces \cite{STONE:2}. Our proof is based on \cite{RUDIN:1} (and also \cite{PATKOWSKA:74}). We prove first auxiliary fact that every open cover of any metrizable space has a locally finite open refinement. We show next the main theorem that every metrizable space is paracompact. The remaining material is devoted to concepts and certain properties needed for the formulation and the proof of that theorem (see also \cite{PCOMPS_1.ABS}).

- Toshihiko Watanabe.
The Brouwer Fixed Point Theorem for Intervals,
Formalized Mathematics 3(1), pages 85-88, 1992. MML Identifier: TREAL_1

**Summary**: The aim is to prove, using Mizar System, the following simplest version of the Brouwer Fixed Point Theorem \cite{BROUWER:1}. {\em For every continuous mapping $f : {\Bbb I} \rightarrow {\Bbb I}$ of the topological unit interval $\Bbb I$ there exists a point $x$ such that $f(x) = x$} (see e.g. \cite{DUG-GRAN:1}, \cite{BROWN:1}).

- Grzegorz Bancerek.
On Powers of Cardinals,
Formalized Mathematics 3(1), pages 89-93, 1992. MML Identifier: CARD_5

**Summary**: In the first section the results of \cite[axiom (30)]{AXIOMS.ABS}\footnote {Axiom (30)\quad ---\quad $n = \{k\in{\Bbb N}: k < n\}$ for every natural number $n$.}, i.e. the correspondence between natural and ordinal (cardinal) numbers are shown. The next section is concerned with the concepts of infinity and cofinality (see \cite{ZFREFLE1.ABS}), and introduces alephs as infinite cardinal numbers. The arithmetics of alephs, i.e. some facts about addition and multiplication, is present in the third section. The concepts of regular and irregular alephs are introduced in the fourth section, and the fact that $\aleph_0$ and every non-limit cardinal number are regular is proved there. Finally, for every alephs $\alpha$ and $\beta$ $$\alpha^\beta = \left\{ \begin{array}{ll} 2^\beta,& {\rm if}\ \alpha\leq\beta,\\ \sum_{\gamma<\alpha}\gamma^\beta,& {\rm if}\ \beta < {\rm cf}\alpha\ {\rm and} \ \alpha\ {\rm is\ limit\ cardinal},\\ \left(\sum_{\gamma<\alpha}\gamma^\beta\right)^{\rm cf\alpha},& {\rm if\ cf}\alpha \leq \beta \leq \alpha.\\ \end{array}\right.$$ \\ Some proofs are based on \cite{GUZ-ZBIER:1}.

- Yatsuka Nakamura, Jaroslaw Kotowicz.
Basic Properties of Connecting Points with Line Segments in $\calE^2_\rmT$,
Formalized Mathematics 3(1), pages 95-99, 1992. MML Identifier: TOPREAL3

**Summary**: Some properties of line segments in 2-dimensional Euclidean space and some relations between line segments and balls are proved.

- Yatsuka Nakamura, Jaroslaw Kotowicz.
Connectedness Conditions Using Polygonal Arcs,
Formalized Mathematics 3(1), pages 101-106, 1992. MML Identifier: TOPREAL4

**Summary**: A concept of special polygonal arc joining two different points is defined. Any two points in a ball can be connected by this kind of arc, and that is also true for any region in ${\cal E}^2_{\rm T}$.

- Jaroslaw Kotowicz, Yatsuka Nakamura.
Introduction to Go-Board -- Part I,
Formalized Mathematics 3(1), pages 107-115, 1992. MML Identifier: GOBOARD1

**Summary**: In the article we introduce Go-board as some kinds of matrix which elements belong to topological space ${\cal E}^2_{\rm T}$. We define the functor of delaying column in Go-board and relation between Go-board and finite sequence of point from ${\cal E}^2_{\rm T}$. Basic facts about those notations are proved. The concept of the article is based on \cite{TAKE-NAKA}.

- Jaroslaw Kotowicz, Yatsuka Nakamura.
Introduction to Go-Board -- Part II,
Formalized Mathematics 3(1), pages 117-121, 1992. MML Identifier: GOBOARD2

**Summary**: In article we define Go-board determined by finite sequence of points from topological space ${\cal E}^2_{\rm T}$. A few facts about this notation are proved.

- Jaroslaw Kotowicz, Yatsuka Nakamura.
Properties of Go-Board -- Part III,
Formalized Mathematics 3(1), pages 123-124, 1992. MML Identifier: GOBOARD3

**Summary**: Two useful facts about Go-board are proved.

- Jaroslaw Kotowicz, Yatsuka Nakamura.
Go-Board Theorem,
Formalized Mathematics 3(1), pages 125-129, 1992. MML Identifier: GOBOARD4

**Summary**: We prove the Go-board theorem which is a special case of Hex Theorem. The article is based on \cite{TAKE-NAKA}.

- Waldemar Korczynski.
Some Properties of Binary Relations,
Formalized Mathematics 3(1), pages 131-134, 1992. MML Identifier: SYSREL

**Summary**: The article contains some theorems on binary relations, which are used in papers \cite{FF_SIEC.ABS}, \cite{E_SIEC.ABS}, \cite{S_SIEC.ABS}, and other.

- Zbigniew Karno.
Continuity of Mappings over the Union of Subspaces,
Formalized Mathematics 3(1), pages 1-16, 1992. MML Identifier: TMAP_1
- Number 2
- Yatsuka Nakamura, Jaroslaw Kotowicz.
The Jordan's Property for Certain Subsets of the Plane,
Formalized Mathematics 3(2), pages 137-142, 1992. MML Identifier: JORDAN1

**Summary**: Let $S$ be a subset of the topological Euclidean plane ${\cal E}^2_{\rm T}$. We say that $S$ has Jordan's property if there exist two non-empty, disjoint and connected subsets $G_1$ and $G_2$ of ${\cal E}^2_{\rm T}$ such that $S \mathclose{^{\rm c}} = G_1 \cup G_2$ and $\overline{G_1} \setminus G_1 = \overline{G_2} \setminus{G_2}$ (see \cite{TAKE-NAKA}, \cite{Dick}). The aim is to prove that the boundaries of some special polygons in ${\cal E}^2_{\rm T}$ have this property (see Section 3). Moreover, it is proved that both the interior and the exterior of the boundary of any rectangle in ${\cal E}^2_{\rm T}$ is open and connected.

- Zbigniew Karno.
The Lattice of Domains of an Extremally Disconnected Space,
Formalized Mathematics 3(2), pages 143-149, 1992. MML Identifier: TDLAT_3

**Summary**: Let $X$ be a topological space and let $A$ be a subset of $X$. Recall that $A$ is said to be a {\em domain}\/ in $X$ provided ${\rm Int}\, \overline{A} \subseteq A \subseteq \overline{{\rm Int}\,A}$ (see \cite{TOPS_1.ABS}, \cite{ISOMICHI}). Recall also that $A$ is said to be a(n) {\em closed}\/ ({\em open})\/ {\em domain}\/ in $X$ if $A = \overline{{\rm Int}\,A}$ ($A = {\rm Int}\,\overline{A}$, resp.) (see e.g. \cite{KURAT:2}, \cite{TOPS_1.ABS}). It is well-known that for a given topological space all its closed domains form a Boolean lattice, and similarly all its open domains form a Boolean lattice, too (see e.g., \cite{MOST-KURAT:3}, \cite{BIRKHOFF:1}). In \cite{TDLAT_1.ABS} it is proved that all domains of a given topological space form a complemented lattice. One may ask whether the lattice of all domains is Boolean. The aim is to give an answer to this question.\par To present the main results we first recall the definition of a class of topological spaces which is important here. $X$ is called {\em extremally disconnected}\/ if for every open subset $A$ of $X$ the closure $\overline {A}$ is open in $X$ \cite{STONE:1} (comp. \cite{ENGEL:1}). It is shown here, using Mizar System, that {\em the lattice of all domains of a topological space $X$ is modular iff $X$ is extremally disconnected.} Moreover, for every extremally disconnected space the lattice of all its domains coincides with both the lattice of all its closed domains and the lattice of all its open domains. From these facts it follows that {\em the lattice of all domains of a topological space $X$ is Boolean iff $X$ is extremally disconnected.}\par Note that we also review some of the standard facts on discrete, anti-discrete, almost discrete, extremally disconnected and hereditarily extremally disconnected topological spaces (comp. \cite{KURAT:2}, \cite{ENGEL:1}).

- Yatsuka Nakamura, Andrzej Trybulec.
A Mathematical Model of CPU,
Formalized Mathematics 3(2), pages 151-160, 1992. MML Identifier: AMI_1

**Summary**: This paper is based on a previous work of the first author \cite{NAKAMURA1} in which a mathematical model of the computer has been presented. The model deals with random access memory, such as RASP of C. C. Elgot and A. Robinson \cite{ELGOT-ROBIN}, however, it allows for a more realistic modeling of real computers. This new model of computers has been named by the author (Y. Nakamura, \cite{NAKAMURA1}) Architecture Model for Instructions (AMI). It is more developed than previous models, both in the description of hardware (e.g., the concept of the program counter, the structure of memory) as well as in the description of instructions (instruction codes, addresses). The structure of AMI over an arbitrary collection of mathematical domains N consists of: \begin{description} \item{ - }a non-empty set of objects, \item{ - }the instruction counter, \item{ - }a non-empty set of objects called instruction locations, \item{ - }a non-empty set of instruction codes, \item{ - }an instruction code for halting, \item{ - }a set of instructions that are ordered pairs with the first element being an instruction code and the second a finite sequence in which members are either objects of the AMI or elements of one of the domains included in N, \item{ - }a function that assigns to every object of AMI its kind that is either {\em an instruction} or {\em an instruction location} or an element of N, \item{ - }a function that assigns to every instruction its execution that is again a function mapping states of AMI into the set of states. \end{description} By a state of AMI we mean a function that assigns to every object of AMI an element of the same kind. In this paper we develop the theory of AMI. Some properties of AMI are introduced ensuring it to have some properties of real computers: \begin{description} \item{ - }a von Neumann AMI, in which only addresses to instruction locations are stored in the program counter, \item{ - }data oriented, those in which instructions cannot be stored in data locations, \item{ - }halting, in which the execution of the halt instruction is the identity mapping of the states of an AMI, \item{ - }steady programmed, the condition in which the contents of the instruction locations do not change during execution, \item{ - }definite, a property in which only instructions may be stored in instruction locations. \end{description} We present an example of AMI called a Small Concrete Model which has been constructed in \cite{NAKAMURA1}. The Small Concrete Model has only one kind of data: integers and a set of instructions, small but sufficient to cope with integers.

- Czeslaw Bylinski.
Cartesian Categories,
Formalized Mathematics 3(2), pages 161-169, 1992. MML Identifier: CAT_4

**Summary**: We define and prove some simple facts on Cartesian categories and its duals co-Cartesian categories. The Cartesian category is defined as a category with the fixed terminal object, the fixed projections, and the binary products. Category $C$ has finite products if and only if $C$ has a terminal object and for every pair $a, b$ of objects of $C$ the product $a\times b$ exists. We say that a category $C$ has a finite product if every finite family of objects of $C$ has a product. Our work is based on ideas of \cite{SZABO}, where the algebraic properties of the proof theory are investigated. The terminal object of a Cartesian category $C$ is denoted by $\hbox{\bf 1}_C$. The binary product of $a$ and $b$ is written as $a\times b$. The projections of the product are written as $pr_1(a,b)$ and as $pr_2(a,b)$. We define the products $f\times g$ of arrows $f: a\rightarrow a'$ and $g: b\rightarrow b'$ as $:a\times b\rightarrow a'\times b'$.\par Co-Cartesian category is defined dually to the Cartesian category. Dual to a terminal object is an initial object, and to products are coproducts. The initial object of a Cartesian category $C$ is written as $\hbox{\bf 0}_C$. Binary coproduct of $a$ and $b$ is written as $a+b$. Injections of the coproduct are written as $in_1(a,b)$ and as $in_2(a,b)$.

- Hiroshi Yamazaki, Yasunari Shidama.
Algebra of Vector Functions,
Formalized Mathematics 3(2), pages 171-175, 1992. MML Identifier: VFUNCT_1

**Summary**: We develop the algebra of partial vector functions, with domains being algebra of vector functions.

- Zbigniew Karno.
On a Duality between Weakly Separated Subspaces of Topological Spaces,
Formalized Mathematics 3(2), pages 177-182, 1992. MML Identifier: TSEP_2

**Summary**: Let $X$ be a topological space and let $X_{1}$ and $X_{2}$ be subspaces of $X$ with the carriers $A_{1}$ and $A_{2}$, respectively. Recall that $X_{1}$ and $X_{2}$ are {\em weakly separated}\/ if $A_{1} \setminus A_{2}$ and $A_{2} \setminus A_{1}$ are separated (see \cite{TSEP_1.ABS} and also \cite{TMAP_1.ABS} for applications). Our purpose is to list a number of properties of such subspaces, supplementary to those given in \cite{TSEP_1.ABS}. Note that in the Mizar formalism the carrier of any topological space (hence the carrier of any its subspace) is always non--empty, therefore for convenience we list beforehand analogous properties of weakly separated subsets without any additional conditions.\par To present the main results we first formulate a useful definition. We say that $X_{1}$ and $X_{2}$ {\em constitute a decomposition}\/ of $X$ if $A_{1}$ and $A_{2}$ are disjoint and the union of $A_{1}$ and $A_{2}$ covers the carrier of $X$ (comp. \cite{KURAT:2}). We are ready now to present the following duality property between pairs of weakly separated subspaces~: {\em If each pair of subspaces $X_{1}$, $Y_{1}$ and $X_{2}$, $Y_{2}$ of $X$ constitutes a decomposition of $X$, then $X_{1}$ and $X_{2}$ are weakly separated iff $Y_{1}$ and $Y_{2}$ are weakly separated}. From this theorem we get immediately that under the same hypothesis, {\em $X_{1}$ and $X_{2}$ are separated iff $X_{1}$ misses $X_{2}$ and $Y_{1}$ and $Y_{2}$ are weakly separated}. Moreover, we show the following enlargement theorem~: {\em If $X_{i}$ and $Y_{i}$ are subspaces of $X$ such that $Y_{i}$ is a subspace of $X_{i}$ and $Y_{1} \cup Y_{2} = X_{1} \cup X_{2}$ and if $Y_{1}$ and $Y_{2}$ are weakly separated, then $X_{1}$ and $X_{2}$ are weakly separated}. We show also the following dual extenuation theorem~: {\em If $X_{i}$ and $Y_{i}$ are subspaces of $X$ such that $Y_{i}$ is a subspace of $X_{i}$ and $Y_{1} \cap Y_{2} = X_{1} \cap X_{2}$ and if $X_{1}$ and $X_{2}$ are weakly separated, then $Y_{1}$ and $Y_{2}$ are weakly separated}. At the end we give a few properties of weakly separated subspaces in subspaces.

- Pauline N. Kawamoto, Yasushi Fuwa, Yatsuka Nakamura.
Basic Petri Net Concepts,
Formalized Mathematics 3(2), pages 183-187, 1992. MML Identifier: PETRI

**Summary**: This article presents the basic place/transition net structure definition for building various types of Petri nets. The basic net structure fields include places, transitions, and arcs (place-transition, transition-place) which may be supplemented with other fields (e.g., capacity, weight, marking, etc.) as needed. The theorems included in this article are divided into the following categories: deadlocks, traps, and dual net theorems. Here, a dual net is taken as the result of inverting all arcs (place-transition arcs to transition-place arcs and vice-versa) in the original net.

- Hiroshi Imura, Masayoshi Eguchi.
Finite Topological Spaces,
Formalized Mathematics 3(2), pages 189-193, 1992. MML Identifier: FIN_TOPO

**Summary**: By borrowing the concept of neighbourhood from the theory of topological space in continuous cases and extending it to a discrete case such as a space of lattice points we have defined such concepts as boundaries, closures, interiors, isolated points, and connected points as in the case of continuity. We have proved various properties which are satisfied by these concepts.

- Grzegorz Bancerek.
Sets and Functions of Trees and Joining Operations of Trees,
Formalized Mathematics 3(2), pages 195-204, 1992. MML Identifier: TREES_3

**Summary**: In the article we deal with sets of trees and functions yielding trees. So, we introduce the sets of all trees, all finite trees and of all trees decorated by elements from some set. Next, the functions and the finite sequences yielding (finite, decorated) trees are introduced. There are shown some convenient but technical lemmas and clusters concerning with those concepts. In the fourth section we deal with trees decorated by Cartesian product and we introduce the concept of a tree called a substitution of structure of some finite tree. Finally, we introduce the operations of joining trees, i.e. for the finite sequence of trees we define the tree which is made by joining the trees from the sequence by common root. For one and two trees there are introduced the same operations.

- Katarzyna Zawadzka.
The Sum and Product of Finite Sequences of Elements of a Field,
Formalized Mathematics 3(2), pages 205-211, 1992. MML Identifier: FVSUM_1

**Summary**: This article is concerned with a generalization of concepts introduced in \cite{RVSUM_1.ABS}, i.e., there are introduced the sum and the product of finite number of elements of any field. Moreover, the product of vectors which yields a vector is introduced. According to \cite{RVSUM_1.ABS}, some operations on $i$-tuples of elements of field are introduced: addition, subtraction, and complement. Some properties of the sum and the product of finite number of elements of a field are present.

- Grzegorz Bancerek.
Monoids,
Formalized Mathematics 3(2), pages 213-225, 1992. MML Identifier: MONOID_0

**Summary**: The goal of the article is to define the concept of monoid. In the preliminary section we introduce the notion of some properties of binary operations. The second section is concerning with structures with a set and a binary operation on this set: there is introduced the notion corresponding to the notion of some properties of binary operations and there are shown some useful clusters. Next, we are concerning with the structure with a set, a binary operation on the set and with an element of the set. Such a structure is called monoid iff the operation is associative and the element is a unity of the operation. In the fourth section the concept of subsystems of monoid (group) is introduced. Subsystems are submonoids (subgroups) or other parts of monoid (group) with are closed w.r.t. the operation. There are presented facts on inheritness of some properties by subsystems. Finally, there are constructed the examples of groups and monoids: the group $\rangle{\Bbb R},+\langle$ of real numbers with addition, the group ${\Bbb Z}^+$ of integers as the subsystem of the group $\rangle{\Bbb R},+\langle$, the semigroup $\rangle{\Bbb N},+\langle$ of natural numbers as the subsystem of ${\Bbb Z}^+$, and the monoid $\rangle{\Bbb N},+,0\langle$ of natural numbers with addition and zero as monoidal extension of the semigroup $\rangle{\Bbb N},+\langle$. The semigroups of real and natural numbers with multiplication are also introduced. The monoid of finite sequences over some set with concatenation as binary operation and with empty sequence as neutral element is defined in sixth section. Last section deals with monoids with the composition of functions as the operation, i.e. with the monoid of partial and total functions and the monoid of permutations.

- Grzegorz Bancerek.
Monoid of Multisets and Subsets,
Formalized Mathematics 3(2), pages 227-233, 1992. MML Identifier: MONOID_1

**Summary**: The monoid of functions yielding elements of a group is introduced. The monoid of multisets over a set is constructed as such monoid where the target group is the group of natural numbers with addition. Moreover, the generalization of group operation onto the operation on subsets is present. That generalization is used to introduce the group $2^G$ of subsets of a group $G$.

- Anna Lango, Grzegorz Bancerek.
Product of Families of Groups and Vector Spaces,
Formalized Mathematics 3(2), pages 235-240, 1992. MML Identifier: PRVECT_1

**Summary**: In the first section we present properties of fields and Abelian groups in terms of commutativity, associativity, etc. Next, we are concerned with operations on $n$-tuples on some set which are generalization of operations on this set. It is used in third section to introduce the $n$-power of a group and the $n$-power of a field. Besides, we introduce a concept of indexed family of binary (unary) operations over some indexed family of sets and a product of such families which is binary (unary) operation on a product of family sets. We use that product in the last section to introduce the product of a finite sequence of Abelian groups.

- Yatsuka Nakamura, Andrzej Trybulec.
On a Mathematical Model of Programs,
Formalized Mathematics 3(2), pages 241-250, 1992. MML Identifier: AMI_2

**Summary**: We continue the work on mathematical modeling of hardware and software started in \cite{AMI_1.ABS}. The main objective of this paper is the definition of a program. We start with the concept of partial product, i.e. the set of all partial functions $f$ from $I$ to $\bigcup_{i\in I} A_i$, fulfilling the condition $f.i \in A_i$ for $i \in dom f$. The computation and the result of a computation are defined in usual way. A finite partial state is called autonomic if the result of a computation starting with it does not depend on the remaining memory and an AMI is called programmable if it has a non empty autonomic partial finite state. We prove the consistency of the following set of properties of an AMI: data-oriented, halting, steady-programmed, realistic and programmable. For this purpose we define a trivial AMI. It has only the instruction counter and one instruction location. The only instruction of it is the halt instruction. A preprogram is a finite partial state that halts. We conclude with the definition of a program of a partial function $F$ mapping the set of the finite partial states into itself. It is a finite partial state $s$ such that for every finite partial state $s' \in dom F$ the result of any computation starting with $s+s'$ includes $F.s'$.

- Jaroslaw Kotowicz, Beata Madras, Malgorzata Korolkiewicz.
Basic Notation of Universal Algebra,
Formalized Mathematics 3(2), pages 251-253, 1992. MML Identifier: UNIALG_1

**Summary**: We present the basic notation of universal algebra.

- Jaroslaw Kotowicz, Konrad Raczkowski.
Coherent Space,
Formalized Mathematics 3(2), pages 255-261, 1992. MML Identifier: COH_SP

**Summary**: Coherent Space, web of coherent space and two categories: category of coherent spaces and category of tolerances on same fixed set.

- Jozef Bialas.
Properties of the Intervals of Real Numbers,
Formalized Mathematics 3(2), pages 263-269, 1992. MML Identifier: MEASURE5

**Summary**: The paper contains definitions and basic properties of the intervals of real numbers.\par The article includes the text being a continuation of the paper \cite{MEASURE4.ABS}. Some theorems concerning basic properties of intervals are proved.

- Wojciech A. Trybulec.
Subspaces of Real Linear Space generated by One, Two, or Three Vectors and Their Cosets,
Formalized Mathematics 3(2), pages 271-274, 1992. MML Identifier: RLVECT_4

**Summary**:

- Jaroslaw Kotowicz.
Functions and Finite Sequences of Real Numbers,
Formalized Mathematics 3(2), pages 275-278, 1992. MML Identifier: RFINSEQ

**Summary**: We define notions of fiberwise equipotent functions, non-increasing finite sequences of real numbers and new operations on finite sequences. Equivalent conditions for fiberwise equivalent functions and basic facts about new constructions are shown.

- Jaroslaw Kotowicz, Yuji Sakai.
Properties of Partial Functions from a Domain to the Set of Real Numbers,
Formalized Mathematics 3(2), pages 279-288, 1992. MML Identifier: RFUNCT_3

**Summary**: The article consists of two parts. In the first one we consider notion of nonnegative and nonpositive part of a real numbers. In the second we consider partial function from a domain to the set of real numbers (or more general to a domain). We define a few new operations for these functions and show connections between finite sequences of real numbers and functions which domain is finite. We introduce {\em integrations} for finite domain real valued functions.

- Michal Muzalewski.
Domains of Submodules, Join and Meet of Finite Sequences of Submodules and Quotient Modules,
Formalized Mathematics 3(2), pages 289-296, 1992. MML Identifier: LMOD_7

**Summary**: Notions of domains of submodules, join and meet of finite sequences of submodules and quotient modules. A few basic theorems and schemes related to these notions are proved.

- Zbigniew Karno.
Remarks on Special Subsets of Topological Spaces,
Formalized Mathematics 3(2), pages 297-303, 1992. MML Identifier: TOPS_3

**Summary**: Let $X$ be a topological space and let $A$ be a subset of $X$. Recall that $A$ is {\em nowhere dense}\/ in $X$ if its closure is a boundary subset of $X$, i.e., if ${\rm Int}\,\overline{A} = \emptyset$ (see \cite{KURAT:2}). We introduce here the concept of everywhere dense subsets in $X$, which is dual to the above one. Namely, $A$ is said to be {\em everywhere dense}\/ in $X$ if its interior is a dense subset of $X$, i.e., if $\overline{{\rm Int}\,A} =$ the carrier of $X$.\par Our purpose is to list a number of properties of such sets (comp. \cite{TOPS_1.ABS}). As a sample we formulate their two dual characterizations. The first one characterizes thin sets in $X$~: {\em $A$ is nowhere dense iff for every open nonempty subset $G$ of $X$ there is an open nonempty subset of $X$ contained in $G$ and disjoint from $A$}. The corresponding second one characterizes thick sets in $X$~: {\em $A$ is everywhere dense iff for every closed subset $F$ of $X$ distinct from the carrier of $X$ there is a closed subset of $X$ distinct from the carrier of $X$, which contains $F$ and together with $A$ covers the carrier of $X$}. We also give some connections between both these concepts. Of course, {\em $A$ is everywhere (nowhere) dense in $X$ iff its complement is nowhere (everywhere) dense}. Moreover, {\em $A$ is nowhere dense iff there are two subsets of $X$, $C$ boundary closed and $B$ everywhere dense, such that $A = C \cap B$ and $C \cup B$ covers the carrier of $X$}. Dually, {\em $A$ is everywhere dense iff there are two disjoint subsets of $X$, $C$ open dense and $B$ nowhere dense, such that $A = C \cup B$}.\par Note that some relationships between everywhere (nowhere) dense sets in $X$ and everywhere (nowhere) dense sets in subspaces of $X$ are also indicated.

- Zbigniew Karno.
On Discrete and Almost Discrete Topological Spaces,
Formalized Mathematics 3(2), pages 305-310, 1992. MML Identifier: TEX_1

**Summary**: A topological space $X$ is called {\em almost discrete}\/ if every open subset of $X$ is closed; equivalently, if every closed subset of $X$ is open (comp. \cite{KURAT:2}, \cite{KURAT:3}). Almost discrete spaces were investigated in Mizar formalism in \cite{TDLAT_3.ABS}. We present here a few properties of such spaces supplementary to those given in \cite{TDLAT_3.ABS}.\par Most interesting is the following characterization~: {\em A topological space $X$ is almost discrete iff every nonempty subset of $X$ is not nowhere dense}. Hence, {\em $X$ is non almost discrete iff there is an everywhere dense subset of $X$ different from the carrier of $X$}. We have an analogous characterization of discrete spaces~: {\em A topological space $X$ is discrete iff every nonempty subset of $X$ is not boundary}. Hence, {\em $X$ is non discrete iff there is a dense subset of $X$ different from the carrier of $X$}. It is well known that the class of all almost discrete spaces contains both the class of discrete spaces and the class of anti-discrete spaces (see e.g., \cite{TDLAT_3.ABS}). Observations presented here show that the class of all almost discrete non discrete spaces is not contained in the class of anti-discrete spaces and the class of all almost discrete non anti-discrete spaces is not contained in the class of discrete spaces. Moreover, the class of almost discrete non discrete non anti-discrete spaces is nonempty. To analyse these interdependencies we use various examples of topological spaces constructed here in Mizar formalism.

- Yatsuka Nakamura, Jaroslaw Kotowicz.
The Jordan's Property for Certain Subsets of the Plane,
Formalized Mathematics 3(2), pages 137-142, 1992. MML Identifier: JORDAN1
**Volume 4, 1993**- Number 1
- Katarzyna Zawadzka.
The Product and the Determinant of Matrices with Entries in a Field,
Formalized Mathematics 4(1), pages 1-8, 1993. MML Identifier: MATRIX_3

**Summary**: Concerned with a generalization of concepts introduced in \cite{MATRIX_1.ABS}, i.e. there are introduced the sum and the product of matrices of any dimension of elements of any field.

- Yuji Sakai, Jaroslaw Kotowicz.
Introduction to Theory of Rearrangement,
Formalized Mathematics 4(1), pages 9-13, 1993. MML Identifier: REARRAN1

**Summary**: An introduction to the rearrangement theory for finite functions (e.g. with the finite domain and codomain). The notion of generators and cogenerators of finite sets (equivalent to the order in the language of finite sequences) has been defined. The notion of rearrangement for a function into finite set is presented. Some basic properties of these notions have been proved.

- Andrzej Trybulec.
Many-sorted Sets,
Formalized Mathematics 4(1), pages 15-22, 1993. MML Identifier: PBOOLE

**Summary**: The article deals with parameterized families of sets. When treated in a similar way as sets (due to systematic overloading notation used for sets) they are called many sorted sets. For instance, if $x$ and $X$ are two many-sorted sets (with the same set of indices $I$) then relation $x \in X$ is defined as $\forall_{i \in I} x_i \in X_i$.\par I was prompted by a remark in a paper by Tarlecki and Wirsing: ``Throughout the paper we deal with many-sorted sets, functions, relations etc. ... We feel free to use any standard set-theoretic notation without explicit use of indices'' \cite[p.~97]{Tar-Wir1}. The aim of this work was to check the feasibility of such approach in Mizar. It works.\par Let us observe some peculiarities: \begin{itemize} \item[-] empty set (i.e. the many sorted set with empty set of indices) belongs to itself (theorem 133), \item[-] we get two different inclusions $X \subseteq Y$ iff $\forall_{i \in I} X_i \subseteq Y_i$ and $X \sqsubseteq Y$ iff $\forall_x x \in X \Rightarrow x \in Y$ equivalent only for sets that yield non empty values. \end{itemize} Therefore the care is advised.

- Ewa Burakowska.
Subalgebras of the Universal Algebra. Lattices of Subalgebras,
Formalized Mathematics 4(1), pages 23-27, 1993. MML Identifier: UNIALG_2

**Summary**: Introduces a definition of a subalgebra of a universal algebra. A notion of similar algebras and basic operations on subalgebras such as a subalgebra generated by a set, the intersection and the sum of two subalgebras were introduced. Some basic facts concerning the above notions have been proved. The article also contains the definition of a lattice of subalgebras of a universal algebra.

- Bogdan Nowak, Andrzej Trybulec.
Hahn-Banach Theorem,
Formalized Mathematics 4(1), pages 29-34, 1993. MML Identifier: HAHNBAN

**Summary**: We prove a version of Hahn-Banach Theorem.

- Jolanta Kamienska, Jaroslaw Stanislaw Walijewski.
Homomorphisms of Lattices, Finite Join and Finite Meet,
Formalized Mathematics 4(1), pages 35-40, 1993. MML Identifier: LATTICE4

**Summary**:

- Jolanta Kamienska.
Representation Theorem for Heyting Lattices,
Formalized Mathematics 4(1), pages 41-45, 1993. MML Identifier: OPENLATT

**Summary**:

- Jaroslaw Stanislaw Walijewski.
Representation Theorem for Boolean Algebras,
Formalized Mathematics 4(1), pages 45-50, 1993. MML Identifier: LOPCLSET

**Summary**:

- Andrzej Trybulec, Yatsuka Nakamura.
Some Remarks on the Simple Concrete Model of Computer,
Formalized Mathematics 4(1), pages 51-56, 1993. MML Identifier: AMI_3

**Summary**: We prove some results on {\bf SCM} needed for the proof of the correctness of Euclid's algorithm. We introduce the following concepts: \begin{itemize} \item[-] starting finite partial state (Start-At$(l)$), then assigns to the instruction counter an instruction location (and consists only of this assignment), \item[-] programmed finite partial state, that consists of the instructions (to be more precise, a finite partial state with the domain consisting of instruction locations). \end{itemize} We define for a total state $s$ what it means that $s$ starts at $l$ (the value of the instruction counter in the state $s$ is $l$) and $s$ halts at $l$ (the halt instruction is assigned to $l$ in the state $s$). Similar notions are defined for finite partial states.

- Andrzej Trybulec, Yatsuka Nakamura.
Euclid's Algorithm,
Formalized Mathematics 4(1), pages 57-60, 1993. MML Identifier: AMI_4

**Summary**: The main goal of the paper is to prove the correctness of the Euclid's algorithm for {\bf SCM}. We define the Euclid's algorithm and describe the natural semantics of it. Eventually we prove that the Euclid's algorithm computes the Euclid's function. Let us observe that the Euclid's function is defined as a function mapping finite partial states to finite partial states of {\bf SCM} rather than pairs of integers to integers.

- Grzegorz Bancerek, Piotr Rudnicki.
Development of Terminology for \bf SCM,
Formalized Mathematics 4(1), pages 61-67, 1993. MML Identifier: SCM_1

**Summary**: We develop a higher level terminology for the {\bf SCM} machine defined by Nakamura and Trybulec in \cite{AMI_1.ABS}. Among numerous technical definitions and lemmas we define a complexity measure of a halting state of {\bf SCM} and a loader for {\bf SCM} for arbitrary finite sequence of instructions. In order to test the introduced terminology we discuss properties of eight shortest halting programs, one for each instruction.

- Grzegorz Bancerek, Piotr Rudnicki.
Two Programs for \bf SCM. Part I -- Preliminaries,
Formalized Mathematics 4(1), pages 69-72, 1993. MML Identifier: PRE_FF

**Summary**: In two articles (this one and \cite{FIB_FUSC.ABS}) we discuss correctness of two short programs for the {\bf SCM} machine: one computes Fibonacci numbers and the other computes the {\em fusc} function of Dijkstra \cite{DIJKSTRA}. The limitations of current Mizar implementation rendered it impossible to present the correctness proofs for the programs in one article. This part is purely technical and contains a number of very specific lemmas about integer division, floor, exponentiation and logarithms. The formal definitions of the Fibonacci sequence and the {\em fusc} function may be of general interest.

- Grzegorz Bancerek, Piotr Rudnicki.
Two Programs for \bf SCM. Part II -- Programs,
Formalized Mathematics 4(1), pages 73-75, 1993. MML Identifier: FIB_FUSC

**Summary**: We prove the correctness of two short programs for the {\bf SCM} machine: one computes Fibonacci numbers and the other computes the {\em fusc} function of Dijkstra \cite{DIJKSTRA}. The formal definitions of these functions can be found in \cite{PRE_FF.ABS}. We prove the total correctness of the programs in two ways: by conducting inductions on computations and inductions on input data. In addition we characterize the concrete complexity of the programs as defined in \cite{SCM_1.ABS}.

- Grzegorz Bancerek.
Joining of Decorated Trees,
Formalized Mathematics 4(1), pages 77-82, 1993. MML Identifier: TREES_4

**Summary**: This is the continuation of the sequence of articles on trees (see \cite{TREES_1.ABS}, \cite{TREES_2.ABS}, \cite{TREES_3.ABS}). The main goal is to introduce joining operations on decorated trees corresponding with operations introduced in \cite{TREES_3.ABS}. We will also introduce the operation of substitution. In the last section we dealt with trees decorated by Cartesian product, i.e. we showed some lemmas on joining operations applied to such trees.

- Takaya Nishiyama, Yasuho Mizuhara.
Binary Arithmetics,
Formalized Mathematics 4(1), pages 83-86, 1993. MML Identifier: BINARITH

**Summary**: Formalizes the basic concepts of binary arithmetic and its related operations. We present the definitions for the following logical operators: 'or' and 'xor' (exclusive or) and include in this article some theorems concerning these operators. We also introduce the concept of an $n$-bit register. Such registers are used in the definition of binary unsigned arithmetic presented in this article. Theorems on the relationships of such concepts to the operations of natural numbers are also given.

- Pauline N. Kawamoto, Yasushi Fuwa, Yatsuka Nakamura.
Basic Concepts for Petri Nets with Boolean Markings,
Formalized Mathematics 4(1), pages 87-90, 1993. MML Identifier: BOOLMARK

**Summary**: Contains basic concepts for Petri nets with Boolean markings and the firability$\slash$firing of single transitions as well as sequences of transitions \cite{Nakamura:5}. The concept of a Boolean marking is introduced as a mapping of a Boolean TRUE$\slash$FALSE to each of the places in a place$\slash$transition net. This simplifies the conventional definitions of the firability and firing of a transition. One note of caution in this article - the definition of firing a transition does not require that the transition be firable. Therefore, it is advisable to check that transitions ARE firable before firing them.

- Grzegorz Bancerek, Piotr Rudnicki.
On Defining Functions on Trees,
Formalized Mathematics 4(1), pages 91-101, 1993. MML Identifier: DTCONSTR

**Summary**: The continuation of the sequence of articles on trees (see \cite{TREES_1.ABS}, \cite{TREES_2.ABS}, \cite{TREES_3.ABS}, \cite{TREES_4.ABS}) and on context-free grammars (\cite{LANG1.ABS}). We define the set of complete parse trees for a given context-free grammar. Next we define the scheme of induction for the set and the scheme of defining functions by induction on the set. For each symbol of a context-free grammar we define the terminal, the pretraversal, and the posttraversal languages. The introduced terminology is tested on the example of Peano naturals.

- Beata Madras.
Product of Family of Universal Algebras,
Formalized Mathematics 4(1), pages 103-108, 1993. MML Identifier: PRALG_1

**Summary**: The product of two algebras, trivial algebra determined by an empty set and product of a family of algebras are defined. Some basic properties are shown.

- Malgorzata Korolkiewicz.
Homomorphisms of Algebras. Quotient Universal Algebra,
Formalized Mathematics 4(1), pages 109-113, 1993. MML Identifier: ALG_1

**Summary**: The first part introduces homomorphisms of universal algebras and their basic properties. The second is concerned with the construction of a quotient universal algebra. The first isomorphism theorem is proved.

- Beata Perkowska.
Free Universal Algebra Construction,
Formalized Mathematics 4(1), pages 115-120, 1993. MML Identifier: FREEALG

**Summary**: A construction of the free universal algebra with fixed signature and a given set of generators.

- Agnieszka Banachowicz, Anna Winnicka.
Complex Sequences,
Formalized Mathematics 4(1), pages 121-124, 1993. MML Identifier: COMSEQ_1

**Summary**: Definitions of complex sequence and operations on sequences (multiplication of sequences and multiplication by a complex number, addition, subtraction, division and absolute value of sequence) are given. We followed \cite{SEQ_1.ABS}.

- Zbigniew Karno.
Maximal Discrete Subspaces of Almost Discrete Topological Spaces,
Formalized Mathematics 4(1), pages 125-135, 1993. MML Identifier: TEX_2

**Summary**: Let $X$ be a topological space and let $D$ be a subset of $X$. $D$ is said to be {\em discrete}\/ provided for every subset $A$ of $X$ such that $A \subseteq D$ there is an open subset $G$ of $X$ such that $A = D \cap G$\/ (comp. e.g., \cite{KURAT:2}). A discrete subset $M$ of $X$ is said to be {\em maximal discrete}\/ provided for every discrete subset $D$ of $X$ if $M \subseteq D$ then $M = D$. A subspace of $X$ is {\em discrete}\/ ({\em maximal discrete}) iff its carrier is discrete (maximal discrete) in $X$.\par Our purpose is to list a number of properties of discrete and maximal discrete sets in Mizar formalism. In particular, we show here that {\em if $D$ is dense and discrete then $D$ is maximal discrete}; moreover, {\em if $D$ is open and maximal discrete then $D$ is dense}. We discuss also the problem of the existence of maximal discrete subsets in a topological space.\par To present the main results we first recall a definition of a class of topological spaces considered herein. A topological space $X$ is called {\em almost discrete}\/ if every open subset of $X$ is closed; equivalently, if every closed subset of $X$ is open. Such spaces were investigated in Mizar formalism in \cite{TDLAT_3.ABS} and \cite{TEX_1.ABS}. We show here that {\em every almost discrete space contains a maximal discrete subspace and every such subspace is a retract of the enveloping space}. Moreover, {\em if $X_{0}$ is a maximal discrete subspace of an almost discrete space $X$ and $r : X \rightarrow X_{0}$ is a continuous retraction, then $r^{-1}(x) = \overline{\{x\}}$ for every point $x$ of $X$ belonging to $X_{0}$}. This fact is a specialization, in the case of almost discrete spaces, of the theorem of M.H. Stone that every topological space can be made into a $T_{0}$-space by suitable identification of points (see \cite{STONE:3}).

- Zbigniew Karno.
On Nowhere and Everywhere Dense Subspaces of Topological Spaces,
Formalized Mathematics 4(1), pages 137-146, 1993. MML Identifier: TEX_3

**Summary**: Let $X$ be a topological space and let $X_{0}$ be a subspace of $X$ with the carrier $A$. $X_{0}$ is called {\em boundary}\/ ({\em dense}) in $X$ if $A$ is boundary (dense), i.e., ${\rm Int}\,A = \emptyset$ ($\overline{A} =$ the carrier of $X$); $X_{0}$ is called {\em nowhere dense}\/ ({\em everywhere dense}) in $X$ if $A$ is nowhere dense (everywhere dense), i.e., ${\rm Int}\,\overline{A} = \emptyset$ ($\overline{{\rm Int}\,A} =$ the carrier of $X$) (see \cite{TOPS_3.ABS} and comp. \cite{KURAT:2}).\par Our purpose is to list, using Mizar formalism, a number of properties of such subspaces, mostly in non-discrete (non-almost-discrete) spaces (comp. \cite{TOPS_3.ABS}). Recall that $X$ is called {\em discrete}\/ if every subset of $X$ is open (closed); $X$ is called {\em almost discrete}\/ if every open subset of $X$ is closed; equivalently, if every closed subset of $X$ is open (see \cite{TDLAT_3.ABS}, \cite{TEX_1.ABS} and comp. \cite{KURAT:2}, \cite{KURAT:3}). We have the following characterization of non-discrete spaces: {\em $X$ is non-discrete iff there exists a boundary subspace in $X$}. Hence, {\em $X$ is non-discrete iff there exists a dense proper subspace in $X$}. We have the following analogous characterization of non-almost-discrete spaces: {\em $X$ is non-almost-discrete iff there exists a nowhere dense subspace in $X$}. Hence, {\em $X$ is non-almost-discrete iff there exists an everywhere dense proper subspace in $X$}.\par Note that some interdependencies between boundary, dense, nowhere and everywhere dense subspaces are also indicated. These have the form of observations in the text and they correspond to the existential and to the conditional clusters in the Mizar System. These clusters guarantee the existence and ensure the extension of types supported automatically by the Mizar System.

- Katarzyna Zawadzka.
The Product and the Determinant of Matrices with Entries in a Field,
Formalized Mathematics 4(1), pages 1-8, 1993. MML Identifier: MATRIX_3
**Volume 5, 1996**- Number 1
- Yasushi Tanaka.
On the Decomposition of the States of SCM,
Formalized Mathematics 5(1), pages 1-8, 1996. MML Identifier: AMI_5

**Summary**: This article continues the development of the basic terminology for the {\bf SCM} as defined in \cite{AMI_1.ABS}, \cite{AMI_2.ABS}, \cite{AMI_3.ABS}. There is developed of the terminology for discussing static properties of instructions (i.e. not related to execution), for data locations, instruction locations, as well as for states and partial states of {\bf SCM}. The main contribution of the article consists in characterizing {\bf SCM} computations starting in states containing autonomic finite partial states.

- Grzegorz Bancerek, Piotr Rudnicki.
On Defining Functions on Binary Trees,
Formalized Mathematics 5(1), pages 9-13, 1996. MML Identifier: BINTREE1

**Summary**: This article is a continuation of an article on defining functions on trees (see \cite{DTCONSTR.ABS}). In this article we develop terminology specialized for binary trees, first defining binary trees and binary grammars. We recast the induction principle for the set of parse trees of binary grammars and the scheme of defining functions inductively with the set as domain. We conclude with defining the scheme of defining such functions by lambda-like expressions.

- Grzegorz Bancerek, Piotr Rudnicki.
A Compiler of Arithmetic Expressions for SCM,
Formalized Mathematics 5(1), pages 15-20, 1996. MML Identifier: SCM_COMP

**Summary**: We define a set of binary arithmetic expressions with the following operations: $+$, $-$, $\cdot$, {\tt mod}, and {\tt div} and formalize the common meaning of the expressions in the set of integers. Then, we define a compile function that for a given expression results in a program for the {\bf SCM} machine defined by Nakamura and Trybulec in \cite{AMI_1.ABS}. We prove that the generated program when loaded into the machine and executed computes the value of the expression. The program uses additional memory and runs in time linear in length of the expression.

- Jozef Bialas.
Some Properties of the Intervals,
Formalized Mathematics 5(1), pages 21-26, 1996. MML Identifier: MEASURE6

**Summary**:

- Yasuho Mizuhara, Takaya Nishiyama.
Binary Arithmetics, Addition and Subtraction of Integers,
Formalized Mathematics 5(1), pages 27-29, 1996. MML Identifier: BINARI_2

**Summary**: This article is a continuation of \cite{BINARITH.ABS} and presents the concepts of binary arithmetic operations for integers. There is introduced 2's complement representation of integers and natural numbers to integers are expanded. The binary addition and subtraction for integers are defined and theorems on the relationship between binary and numerical operations presented.

- Agnieszka Julia Marasik.
Boolean Properties of Lattices,
Formalized Mathematics 5(1), pages 31-35, 1996. MML Identifier: BOOLEALG

**Summary**:

- Andrzej Trybulec.
Many Sorted Algebras,
Formalized Mathematics 5(1), pages 37-42, 1996. MML Identifier: MSUALG_1

**Summary**: The basic purpose of the paper is to prepare preliminaries of the theory of many sorted algebras. The concept of the signature of a many sorted algebra is introduced as well as the concept of many sorted algebra itself. Some auxiliary related notions are defined. The correspondence between (1 sorted) universal algebras \cite{UNIALG_1.ABS} and many sorted algebras with one sort only is described by introducing two functors mapping one into the other. The construction is done this way that the composition of both functors is the identity on universal algebras.

- Artur Kornilowicz.
On the Group of Inner Automorphisms,
Formalized Mathematics 5(1), pages 43-45, 1996. MML Identifier: AUTGROUP

**Summary**:

- Ewa Burakowska.
Subalgebras of Many Sorted Algebra. Lattice of Subalgebras,
Formalized Mathematics 5(1), pages 47-54, 1996. MML Identifier: MSUALG_2

**Summary**:

- Beata Madras.
Products of Many Sorted Algebras,
Formalized Mathematics 5(1), pages 55-60, 1996. MML Identifier: PRALG_2

**Summary**: Product of two many sorted universal algebras and product of family of many sorted universal algebras are defined in this article. Operations on functions, such that commute, Frege, are also introduced.

- Malgorzata Korolkiewicz.
Homomorphisms of Many Sorted Algebras,
Formalized Mathematics 5(1), pages 61-65, 1996. MML Identifier: MSUALG_3

**Summary**: The aim of this article is to present the definition and some properties of homomorphisms of many sorted algebras. Some auxiliary properties of many sorted functions also have been shown.

- Beata Perkowska.
Free Many Sorted Universal Algebra,
Formalized Mathematics 5(1), pages 67-74, 1996. MML Identifier: MSAFREE

**Summary**:

- Mariusz Zynel, Adam Guzowski.
\Tzero\ Topological Spaces,
Formalized Mathematics 5(1), pages 75-77, 1996. MML Identifier: T_0TOPSP

**Summary**:

- Malgorzata Korolkiewicz.
Many Sorted Quotient Algebra,
Formalized Mathematics 5(1), pages 79-84, 1996. MML Identifier: MSUALG_4

**Summary**: This article introduces the construction of a many sorted quotient algebra. A few preliminary notions such as a many sorted relation, a many sorted equivalence relation, a many sorted congruence and the set of all classes of a many sorted relation are also formulated.

- Grzegorz Bancerek.
Quantales,
Formalized Mathematics 5(1), pages 85-91, 1996. MML Identifier: QUANTAL1

**Summary**: The concepts of Girard quantales (see \cite{GIRARD:TCS50} and \cite{YETTER}) and Blikle nets (see \cite{BLIKLE}) are introduced.

- Agnieszka Sakowicz, Jaroslaw Gryko, Adam Grabowski.
Sequences in $\calE^N_\rmT$,
Formalized Mathematics 5(1), pages 93-96, 1996. MML Identifier: TOPRNS_1

**Summary**:

- Yatsuka Nakamura, Czeslaw Bylinski.
Extremal Properties of Vertices on Special Polygons. Part I,
Formalized Mathematics 5(1), pages 97-102, 1996. MML Identifier: SPPOL_1

**Summary**: First, extremal properties of endpoints of line segments in n-dimensional Euclidean space are discussed. Some topological properties of line segments are also discussed. Secondly, extremal properties of vertices of special polygons which consist of horizontal and vertical line segments in 2-dimensional Euclidean space, are also derived.

- Yasushi Tanaka.
Relocatability,
Formalized Mathematics 5(1), pages 103-108, 1996. MML Identifier: RELOC

**Summary**: This article defines the concept of relocating the program part of a finite partial state of {\bf SCM} (data part stays intact). The relocated program differs from the original program in that all jump instructions are adjusted by the relocation factor and other instructions remain unchanged. The main theorem states that if a program computes a function then the relocated program computes the same function, and vice versa.

- Zbigniew Karno.
Maximal Anti-Discrete Subspaces of Topological Spaces,
Formalized Mathematics 5(1), pages 109-118, 1996. MML Identifier: TEX_4

**Summary**: Let $X$ be a topological space and let $A$ be a subset of $X$. $A$ is said to be {\em anti-discrete}\/ provided for every open subset $G$ of $X$ either $A \cap G = \emptyset$ or $A \subseteq G$; equivalently, for every closed subset $F$ of $X$ either $A \cap F = \emptyset$ or $A \subseteq F$. An anti-discrete subset $M$ of $X$ is said to be {\em maximal anti-discrete}\/ provided for every anti-discrete subset $A$ of $X$ if $M \subseteq A$ then $M = A$. A subspace of $X$ is {\em maximal anti-discrete}\/ iff its carrier is maximal anti-discrete in $X$. The purpose is to list a few properties of maximal anti-discrete sets and subspaces in Mizar formalism.\par It is shown that every $x \in X$ is contained in a unique maximal anti-discrete subset M$(x)$ of $X$, denoted in the text by MaxADSet($x$). Such subset can be defined by $${\rm M}(x) = \bigcap\ \{ S \subseteq X : x \in S,\ {\rm and}\ S \ {\rm is}\ {\rm open}\ {\rm or}\ {\rm closed}\ {\rm in}\ X\}.$$ It has the following remarkable properties: (1) $y \in {\rm M}(x)$ iff ${\rm M}(y) = {\rm M}(x)$, (2) either ${\rm M}(x) \cap {\rm M}(y) = \emptyset$ or ${\rm M}(x) = {\rm M}(y)$, (3) ${\rm M}(x) = {\rm M}(y)$ iff $\overline{\{x\}} = \overline{\{y\}}$, and (4) ${\rm M}(x) \cap {\rm M}(y) = \emptyset$ iff $\overline{\{x\}} \neq \overline{\{y\}}$. It follows from these properties that $\{{\rm M}(x) : x \in X\}$ is the $T_{0}$-partition of $X$ defined by M.H.~Stone in \cite{STONE:3}.\par Moreover, it is shown that the operation M defined on all subsets of $X$ by $${\rm M}(A) = \bigcup\ \{{\rm M}(x) : x \in A\},$$ denoted in the text by MaxADSet($A$), satisfies the Kuratowski closure axioms (see e.g., \cite{KURAT:2}), i.e., (1) ${\rm M}(A \cup B) = {\rm M}(A) \cup {\rm M}(B)$, (2) ${\rm M}(A) = {\rm M}({\rm M}(A))$, (3) $A \subseteq {\rm M}(A)$,and (4) ${\rm M}(\emptyset) = \emptyset$. Note that this operation commutes with the usual closure operation of $X$, and if $A$ is an open (or a closed) subset of $X$, then ${\rm M}(A) = A$.

- Zbigniew Karno.
On Kolmogorov Topological Spaces,
Formalized Mathematics 5(1), pages 119-124, 1996. MML Identifier: TSP_1

**Summary**: Let $X$ be a topological space. $X$ is said to be {\em $T_{0}$-space}\/ (or {\em Kolmogorov space}) provided for every pair of distinct points $x,\,y \in X$ there exists an open subset of $X$ containing exactly one of these points; equivalently, for every pair of distinct points $x,\,y \in X$ there exists a closed subset of $X$ containing exactly one of these points (see \cite{ALEX-HOPF}, \cite{KURAT:2}, \cite{ENGEL:1}).\par The purpose is to list some of the standard facts on Kolmogorov spaces, using Mizar formalism. As a sample we formulate the following characteristics of such spaces: {\em $X$ is a Kolmogorov space iff for every pair of distinct points $x,\,y \in X$ the closures $\overline{\{x\}}$ and $\overline{\{y\}}$ are distinct}.\par There is also reviewed analogous facts on Kolmogorov subspaces of topological spaces. In the presented approach $T_{0}$-subsets are introduced and some of their properties developed.

- Zbigniew Karno.
Maximal Kolmogorov Subspaces of a Topological Space as Stone Retracts of the Ambient Space,
Formalized Mathematics 5(1), pages 125-130, 1996. MML Identifier: TSP_2

**Summary**: Let $X$ be a topological space. $X$ is said to be {\em $T_{0}$-space}\/ (or {\em Kolmogorov space}) provided for every pair of distinct points $x,\,y \in X$ there exists an open subset of $X$ containing exactly one of these points (see \cite{ALEX-HOPF}, \cite{KURAT:2}, \cite{ENGEL:1}). Such spaces and subspaces were investigated in Mizar formalism in \cite{TSP_1.ABS}. A Kolmogorov subspace $X_{0}$ of a topological space $X$ is said to be {\em maximal}\/ provided for every Kolmogorov subspace $Y$ of $X$ if $X_{0}$ is subspace of $Y$ then the topological structures of $Y$ and $X_{0}$ are the same.\par M.H.~Stone proved in \cite{STONE:3} that every topological space can be made into a Kolmogorov space by identifying points with the same closure (see also \cite{THRON:1}). The purpose is to generalize the Stone result, using Mizar System. It is shown here that: (1) {\em in every topological space $X$ there exists a maximal Kolmogorov subspace $X_{0}$ of $X$}, and (2) {\em every maximal Kolmogorov subspace $X_{0}$ of $X$ is a continuous retract of $X$}. Moreover, {\em if $r : X \rightarrow X_{0}$ is a continuous retraction of $X$ onto a maximal Kolmogorov subspace $X_{0}$ of $X$, then $r^{-1}(x) = {\rm MaxADSet}(x)$ for any point $x$ of $X$ belonging to $X_{0}$, where ${\rm MaxADSet}(x)$ is a unique maximal anti-discrete subset of $X$ containing $x$} (see \cite{TEX_4.ABS} for the precise definition of the set ${\rm MaxADSet}(x)$). The retraction $r$ from the last theorem is defined uniquely, and it is denoted in the text by ``Stone-retraction". It has the following two remarkable properties: $r$ is open, i.e., sends open sets in $X$ to open sets in $X_{0}$, and $r$ is closed, i.e., sends closed sets in $X$ to closed sets in $X_{0}$.\par These results may be obtained by the methods described by R.H. Warren in \cite{WARREN:1}.

- Michal Muzalewski.
Projective Planes,
Formalized Mathematics 5(1), pages 131-136, 1996. MML Identifier: PROJPL_1

**Summary**: The line of points $a$, $b$, denoted by $a\cdot b$ and the point of lines $A$, $B$ denoted by $A\cdot B$ are defined. A few basic theorems related to these notions are proved. An inspiration for such approach comes from so called Leibniz program. Let us recall that the Leibniz program is a program of algebraization of geometry using purely geometric notions. Leibniz formulated his program in opposition to algebraization method developed by Descartes.

- Yozo Toda.
The Formalization of Simple Graphs,
Formalized Mathematics 5(1), pages 137-144, 1996. MML Identifier: SGRAPH1

**Summary**: A graph is simple when \begin{itemize} \parskip -1mm \item it is non-directed, \item there is at most one edge between two vertices, \item there is no loop of length one. \end{itemize} A formalization of simple graphs is given from scratch. There is already an article \cite{GRAPH_1.ABS}, dealing with the similar subject. It is not used as a starting-point, because \cite{GRAPH_1.ABS} formalizes directed non-empty graphs. Given a set of vertices, edge is defined as an (unordered) pair of different two vertices and graph as a pair of a set of vertices and a set of edges.\par The following concepts are introduced: \begin{itemize} \parskip -1mm \item simple graph structure, \item the set of all simple graphs, \item equality relation on graphs. \item the notion of degrees of vertices; the number of edges connected to, or the number of adjacent vertices, \item the notion of subgraphs, \item path, cycle, \item complete and bipartite complete graphs, \end{itemize}\par Theorems proved in this articles include: \begin{itemize} \parskip -1mm \item the set of simple graphs satisfies a certain minimality condition, \item equivalence between two notions of degrees. \end{itemize}

- Katarzyna Zawadzka.
Solvable Groups,
Formalized Mathematics 5(1), pages 145-147, 1996. MML Identifier: GRSOLV_1

**Summary**: The concept of solvable group is introduced. Some theorems concerning heirdom of solvability are proved.

- Yasushi Tanaka.
On the Decomposition of the States of SCM,
Formalized Mathematics 5(1), pages 1-8, 1996. MML Identifier: AMI_5
- Number 2
- Grzegorz Bancerek.
Ideals,
Formalized Mathematics 5(2), pages 149-156, 1996. MML Identifier: FILTER_2

**Summary**: The dual concept to filters (see \cite{FILTER_0.ABS}, \cite{FILTER_1.ABS}) i.e. ideals of a lattice is introduced.

- Grzegorz Bancerek.
Categorial Categories and Slice Categories,
Formalized Mathematics 5(2), pages 157-165, 1996. MML Identifier: CAT_5

**Summary**: By categorial categories we mean categories with categories as objects and morphisms of the form $(C_1, C_2, F)$, where $C_1$ and $C_2$ are categories and $F$ is a functor from $C_1$ into $C_2$.

- Yatsuka Nakamura, Piotr Rudnicki, Andrzej Trybulec, Pauline N. Kawamoto.
Preliminaries to Circuits, I,
Formalized Mathematics 5(2), pages 167-172, 1996. MML Identifier: PRE_CIRC

**Summary**: This article is the first in a series of four articles (continued in \cite{MSAFREE2.ABS}, \cite{CIRCUIT1.ABS}, \cite{CIRCUIT2.ABS}) about modelling circuits by many-sorted algebras.\par Here, we introduce some auxiliary notations and prove auxiliary facts about many sorted sets, many sorted functions and trees.

- Miroslava Kaloper , Piotr Rudnicki.
Minimization of finite state machines,
Formalized Mathematics 5(2), pages 173-184, 1996. MML Identifier: FSM_1

**Summary**: We have formalized deterministic finite state machines closely following the textbook \cite{ddq}, pp. 88--119 up to the minimization theorem. In places, we have changed the approach presented in the book as it turned out to be too specific and inconvenient. Our work also revealed several minor mistakes in the book. After defining a structure for an outputless finite state machine, we have derived the structures for the transition assigned output machine (Mealy) and state assigned output machine (Moore). The machines are then proved similar, in the sense that for any Mealy (Moore) machine there exists a Moore (Mealy) machine producing essentially the same response for the same input. The rest of work is then done for Mealy machines. Next, we define equivalence of machines, equivalence and $k$-equivalence of states, and characterize a process of constructing for a given Mealy machine, the machine equivalent to it in which no two states are equivalent. The final, minimization theorem states: \begin{quotation} \noindent {\bf Theorem 4.5:} Let {\bf M}$_1$ and {\bf M}$_2$ be reduced, connected finite-state machines. Then the state graphs of {\bf M}$_1$ and {\bf M}$_2$ are isomorphic if and only if {\bf M}$_1$ and {\bf M}$_2$ are equivalent. \end{quotation} and it is the last theorem in this article.

- Grzegorz Bancerek.
Subtrees,
Formalized Mathematics 5(2), pages 185-190, 1996. MML Identifier: TREES_9

**Summary**: The concepts of root tree, the set of successors of a node in decorated tree and sets of subtrees are introduced.

- Grzegorz Bancerek.
Terms Over Many Sorted Universal Algebra,
Formalized Mathematics 5(2), pages 191-198, 1996. MML Identifier: MSATERM

**Summary**: Pure terms (without constants) over a signature of many sorted universal algebra and terms with constants from algebra are introduced. Facts on evaluation of a term in some valuation are proved.

- Marian Przemski.
On the Decomposition of the Continuity,
Formalized Mathematics 5(2), pages 199-204, 1996. MML Identifier: DECOMP_1

**Summary**: This article is devoted to functions of general topological spaces. A function from $X$ to $Y$ is $A$-continuous if the counterimage of every open set $V$ of $Y$ belongs to $A$, where $A$ is a collection of subsets of $X$. We give the following characteristics of the continuity, called decomposition of continuity: A function $f$ is continuous if and only if it is both $A$-continuous and $B$-continuous.

- Andrzej Trybulec.
A Scheme for Extensions of Homomorphisms of Manysorted Algebras,
Formalized Mathematics 5(2), pages 205-209, 1996. MML Identifier: MSAFREE1

**Summary**: The aim of this work is to provide a bridge between the theory of context-free grammars developed in \cite{LANG1.ABS}, \cite{DTCONSTR.ABS} and universally free manysorted algebras(\cite{MSAFREE.ABS}. The third scheme proved in the article allows to prove that two homomorphisms equal on the set of free generators are equal. The first scheme is a slight modification of the scheme in \cite{DTCONSTR.ABS} and the second is rather technical, but since it was useful for me, perhaps it might be useful for somebody else. The concept of flattening of a many sorted function $F$ between two manysorted sets $A$ and $B$ (with common set of indices $I$) is introduced for $A$ with mutually disjoint components (pairwise disjoint function -- the concept introduced in \cite{PROB_2.ABS}). This is a function on the union of $A$, that is equal to $F$ on every component of $A$. A trivial many sorted algebra over a signature $S$ is defined with sorts being singletons of corresponding sort symbols. It has mutually disjoint sorts.}

- Adam Grabowski.
The Correspondence Between Homomorphisms of Universal Algebra \& Many Sorted Algebra,
Formalized Mathematics 5(2), pages 211-214, 1996. MML Identifier: MSUHOM_1

**Summary**: The aim of the article is to check the compatibility of the homomorphism of universal algebras introduced in \cite{ALG_1.ABS} and the corresponding concept for many sorted algebras introduced in \cite{MSUALG_3.ABS}.

- Yatsuka Nakamura, Piotr Rudnicki, Andrzej Trybulec, Pauline N. Kawamoto.
Preliminaries to Circuits, II,
Formalized Mathematics 5(2), pages 215-220, 1996. MML Identifier: MSAFREE2

**Summary**: This article is the second in a series of four articles (started with \cite{PRE_CIRC.ABS} and continued in \cite{CIRCUIT1.ABS}, \cite{CIRCUIT2.ABS}) about modelling circuits by many sorted algebras.\par First, we introduce some additional terminology for many sorted signatures. The vertices of such signatures are divided into input vertices and inner vertices. A many sorted signature is called {\em circuit like} if each sort is a result sort of at most one operation. Next, we introduce some notions for many sorted algebras and many sorted free algebras. Free envelope of an algebra is a free algebra generated by the sorts of the algebra. Evaluation of an algebra is defined as a homomorphism from the free envelope of the algebra into the algebra. We define depth of elements of free many sorted algebras.\par A many sorted signature is said to be monotonic if every finitely generated algebra over it is locally finite (finite in each sort). Monotonic signatures are used (see \cite{CIRCUIT1.ABS}, \cite{CIRCUIT2.ABS}) in modelling backbones of circuits without directed cycles.

- Artur Kornilowicz.
On the Group of Automorphisms of Universal Algebra \& Many Sorted Algebra,
Formalized Mathematics 5(2), pages 221-226, 1996. MML Identifier: AUTALG_1

**Summary**: The aim of the article is to check the compatibility of the automorphisms of universal algebras introduced in \cite{ALG_1.ABS} and the corresponding concept for many sorted algebras introduced in \cite{MSUALG_3.ABS}.

- Yatsuka Nakamura, Piotr Rudnicki, Andrzej Trybulec, Pauline N. Kawamoto.
Introduction to Circuits, I,
Formalized Mathematics 5(2), pages 227-232, 1996. MML Identifier: CIRCUIT1

**Summary**: This article is the third in a series of four articles (preceded by \cite{PRE_CIRC.ABS}, \cite{MSAFREE2.ABS} and continued in \cite{CIRCUIT2.ABS}) about modelling circuits by many sorted algebras.\par A circuit is defined as a locally-finite algebra over a circuit-like many sorted signature. For circuits we define notions of input function and of circuit state which are later used (see \cite{CIRCUIT2.ABS}) to define circuit computations. For circuits over monotonic signatures we introduce notions of vertex size and vertex depth that characterize certain graph properties of circuit's signature in terms of elements of its free envelope algebra. The depth of a finite circuit is defined as the maximal depth over its vertices.

- Alexander Yu. Shibakov, Andrzej Trybulec.
The Cantor Set,
Formalized Mathematics 5(2), pages 233-236, 1996. MML Identifier: CANTOR_1

**Summary**: The aim of the paper is to define some basic notions of the theory of topological spaces like basis and prebasis, and to prove their simple properties. The definition of the Cantor set is given in terms of countable product of $\{0,1\}$ and a collection of its subsets to serve as a prebasis.

- Oleg Okhotnikov.
Logical Equivalence of Formulae,
Formalized Mathematics 5(2), pages 237-240, 1996. MML Identifier: CQC_THE3

**Summary**:

- Czeslaw Bylinski.
Some Properties of Restrictions of Finite Sequences,
Formalized Mathematics 5(2), pages 241-245, 1996. MML Identifier: FINSEQ_5

**Summary**: The aim of the paper is to define some basic notions of restrictions of finite sequences.

- Czeslaw Bylinski, Yatsuka Nakamura.
Special Polygons,
Formalized Mathematics 5(2), pages 247-252, 1996. MML Identifier: SPPOL_2

**Summary**:

- Jozef Bialas.
The One-Dimensional Lebesgue Measure,
Formalized Mathematics 5(2), pages 253-258, 1996. MML Identifier: MEASURE7

**Summary**: The paper is the crowning of a series of articles written in the Mizar language, being a formalization of notions needed for the description of the one-dimensional Lebesgue measure. The formalization of the notion as classical as the Lebesgue measure determines the powers of the PC Mizar system as a tool for the strict, precise notation and verification of the correctness of deductive theories. Following the successive articles \cite{SUPINF_1.ABS}, \cite{SUPINF_2.ABS}, \cite{MEASURE1.ABS}, \cite{MEASURE6.ABS} constructed so that the final one should include the definition and the basic properties of the Lebesgue measure, we observe one of the paths relatively simple in the sense of the definition, enabling us the formal introduction of this notion. This way, although toilsome, since such is the nature of formal theories, is greatly instructive. It brings home the proper succession of the introduction of the definitions of intermediate notions and points out to those elements of the theory which determine the essence of the complexity of the notion being introduced.\par The paper includes the definition of the $\sigma$-field of Lebesgue measurable sets, the definition of the Lebesgue measure and the basic set of the theorems describing its properties.

- Andrzej Trybulec.
Categories without Uniqueness of \bf cod and \bf dom,
Formalized Mathematics 5(2), pages 259-267, 1996. MML Identifier: ALTCAT_1

**Summary**: Category theory had been formalized in Mizar quite early \cite{CAT_1.ABS}. This had been done closely to the handbook of S. McLane \cite{MACLANE:1}. In this paper we use a different approach. Category is a triple $$\langle O, {\{ \langle o_1,o_2 \rangle \}}_{o_1,o_2 \in O}, {\{ \circ_{o_1,o_2,o_3}\}}_{o_1,o_2,o_3 \in O} \rangle$$ where $\circ_{o_1,o_2,o_3}: \langle o_2,o_3 \rangle \times \langle o_1,o_2 \rangle \to \langle o_1, o_3 \rangle$ that satisfies usual conditions (associativity and the existence of the identities). This approach is closer to the way in which categories are presented in homological algebra (e.g. \cite{BALCERZYK}, pp.58-59). We do not assume that $\langle o_1,o_2 \rangle$'s are mutually disjoint. If $f$ is simultaneously a morphism from $o_1$ to $o_2$ and $o_1'$ to $o_2$ ($o_1 \neq o_1'$) than different compositions are used ($\circ_{o_1,o_2,o_3}$ or $\circ_{o_1',o_2,o_3}$) to compose it with a morphism $g$ from $o_2$ to $o_3$. The operation $g\cdot f$ has actually six arguments (two visible and four hidden: three objects and the category).\par We introduce some simple properties of categories. Perhaps more than necessary. It is partially caused by the formalization. The functional categories are characterized by the following properties: \begin{itemize} \item quasi-functional that means that morphisms are functions (rather meaningless, if it stands alone) \item semi-functional that means that the composition of morphism is the composition of functions, provided they are functions. \item pseudo-functional that means that the composition of morphisms is the composition of functions. \end{itemize} For categories pseudo-functional is just quasi-functional and semi-functional, but we work in a bit more general setting. Similarly the concept of a discrete category is split into two: \begin{itemize} \item quasi-discrete that means that $\langle o_1,o_2 \rangle$ is empty for $o_1 \neq o_2$ and \item pseudo-discrete that means that $\langle o, o \rangle$ is trivial, i.e. consists of the identity only, in a category. \end{itemize}\par We plan to follow Semadeni-Wiweger book \cite{SEMAD}, in the development the category theory in Mizar. However, the beginning is not very close to \cite{SEMAD}, because of the approach adopted and because we work in Tarski-Grothendieck set theory.

- Artur Kornilowicz.
Extensions of Mappings on Generator Set,
Formalized Mathematics 5(2), pages 269-272, 1996. MML Identifier: EXTENS_1

**Summary**: The aim of the article is to prove the fact that if extensions of mappings on generator set are equal then these mappings are equal. The article contains the properties of epimorphisms and monomorphisms between Many Sorted Algebras.

- Yatsuka Nakamura, Piotr Rudnicki, Andrzej Trybulec, Pauline N. Kawamoto.
Introduction to Circuits, II,
Formalized Mathematics 5(2), pages 273-278, 1996. MML Identifier: CIRCUIT2

**Summary**: This article is the last in a series of four articles (preceded by \cite{PRE_CIRC.ABS}, \cite{MSAFREE2.ABS}, \cite{CIRCUIT1.ABS}) about modelling circuits by many sorted algebras.\par The notion of a circuit computation is defined as a sequence of circuit states. For a state of a circuit the next state is given by executing operations at circuit vertices in the current state, according to denotations of the operations. The values at input vertices at each state of a computation are provided by an external sequence of input values. The process of how input values propagate through a circuit is described in terms of a homomorphism of the free envelope algebra of the circuit into itself. We prove that every computation of a circuit over a finite monotonic signature and with constant input values stabilizes after executing the number of steps equal to the depth of the circuit.

- Artur Kornilowicz.
Definitions and Basic Properties of Boolean \& Union of Many Sorted Sets,
Formalized Mathematics 5(2), pages 279-281, 1996. MML Identifier: MBOOLEAN

**Summary**: In the first part of this article I have proved theorems about boolean of many sorted sets which are corresponded to theorems about boolean of sets, whereas the second part of this article contains propositions about union of many sorted sets. Boolean as well as union of many sorted sets are defined as boolean and union on every sorts.

- Yatsuka Nakamura, Grzegorz Bancerek.
Combining of Circuits,
Formalized Mathematics 5(2), pages 283-295, 1996. MML Identifier: CIRCCOMB

**Summary**: We continue the formalisation of circuits started in \cite{PRE_CIRC.ABS}, \cite{MSAFREE2.ABS}, \cite{CIRCUIT1.ABS}, \cite{CIRCUIT2.ABS}. Our goal was to work out the notation of combining circuits which could be employed to prove the properties of real circuits.

- Grzegorz Bancerek.
Ideals,
Formalized Mathematics 5(2), pages 149-156, 1996. MML Identifier: FILTER_2
- Number 3
- Yatsuka Nakamura, Piotr Rudnicki.
Vertex Sequences Induced by Chains,
Formalized Mathematics 5(3), pages 297-304, 1996. MML Identifier: GRAPH_2

**Summary**: In the three preliminary sections to the article we define two operations on finite sequences which seem to be of general interest. The first is the $cut$ operation that extracts a contiguous chunk of a finite sequence from a position to a position. The second operation is a glueing catenation that given two finite sequences catenates them with removal of the first element of the second sequence. The main topic of the article is to define an operation which for a given chain in a graph returns the sequence of vertices through which the chain passes. We define the exact conditions when such an operation is uniquely definable. This is done with the help of the so called two-valued alternating finite sequences. We also prove theorems about the existence of simple chains which are subchains of a given chain. In order to do this we define the notion of a finite subsequence of a typed finite sequence.

- Andrzej Iwaniuk.
On the Lattice of Subspaces of a Vector Space,
Formalized Mathematics 5(3), pages 305-308, 1996. MML Identifier: VECTSP_8

**Summary**:

- Janusz Ganczarski.
On the Lattice of Subgroups of a Group,
Formalized Mathematics 5(3), pages 309-312, 1996. MML Identifier: LATSUBGR

**Summary**:

- Miroslaw Jan Paszek.
On the Lattice of Subalgebras of a Universal Algebra,
Formalized Mathematics 5(3), pages 313-316, 1996. MML Identifier: UNIALG_3

**Summary**:

- Andrzej Trybulec.
On the Decomposition of Finite Sequences,
Formalized Mathematics 5(3), pages 317-322, 1996. MML Identifier: FINSEQ_6

**Summary**:

- Yatsuka Nakamura, Andrzej Trybulec.
Decomposing a Go-Board into Cells,
Formalized Mathematics 5(3), pages 323-328, 1996. MML Identifier: GOBOARD5

**Summary**:

- Grzegorz Bancerek.
Indexed Category,
Formalized Mathematics 5(3), pages 329-337, 1996. MML Identifier: INDEX_1

**Summary**: The concept of indexing of a category (a part of indexed category, see \cite{TarleckiBurstallGoguen}) is introduced as a pair formed by a many sorted category and a many sorted functor. The indexing of a category $C$ against to \cite{TarleckiBurstallGoguen} is not a functor but it can be treated as a functor from $C$ into some categorial category (see \cite{CAT_5.ABS}). The goal of the article is to work out the notation necessary to define institutions (see \cite{GoguenBurstall}).

- Robert Milewski.
Associated Matrix of Linear Map,
Formalized Mathematics 5(3), pages 339-345, 1996. MML Identifier: MATRLIN

**Summary**:

- Andrzej Trybulec.
On the Geometry of a Go-Board,
Formalized Mathematics 5(3), pages 347-352, 1996. MML Identifier: GOBOARD6

**Summary**:

- Jozef Bialas, Yatsuka Nakamura.
The Theorem of Weierstrass,
Formalized Mathematics 5(3), pages 353-359, 1996. MML Identifier: WEIERSTR

**Summary**:

- Jozef Bialas, Yatsuka Nakamura.
Dyadic Numbers and T$_4$ Topological Spaces,
Formalized Mathematics 5(3), pages 361-366, 1996. MML Identifier: URYSOHN1

**Summary**:

- Grzegorz Bancerek, Yatsuka Nakamura.
Full Adder Circuit. Part I,
Formalized Mathematics 5(3), pages 367-380, 1996. MML Identifier: FACIRC_1

**Summary**: We continue the formalisation of circuits started by Piotr Rudnicki, Andrzej Trybulec, Pauline Kawamoto, and the second author in \cite{PRE_CIRC.ABS}, \cite{MSAFREE2.ABS}, \cite{CIRCUIT1.ABS}, \cite{CIRCUIT2.ABS}. The first step in proving properties of full $n$-bit adder circuit, i.e. 1-bit adder, is presented. We employ the notation of combining circuits introduced in \cite{CIRCCOMB.ABS}.

- Grzegorz Bancerek.
Continuous, Stable, and Linear Maps of Coherence Spaces,
Formalized Mathematics 5(3), pages 381-393, 1996. MML Identifier: COHSP_1

**Summary**:

- Artur Kornilowicz.
Some Basic Properties of Many Sorted Sets,
Formalized Mathematics 5(3), pages 395-399, 1996. MML Identifier: PZFMISC1

**Summary**:

- Oleg Okhotnikov.
Replacement of Subtrees in a Tree,
Formalized Mathematics 5(3), pages 401-403, 1996. MML Identifier: TREES_A

**Summary**: This paper is based on previous works \cite{TREES_1.ABS}, \cite{TREES_2.ABS} in which the operation replacement of subtree in a tree has been defined. We extend this notion for arbitrary non empty antichain.

- Grzegorz Bancerek.
Minimal Signature for Partial Algebra,
Formalized Mathematics 5(3), pages 405-414, 1996. MML Identifier: PUA2MSS1

**Summary**: The concept of characterizing of partial algebras by many sorted signature is introduced, i.e. we say that a signature $S$ characterizes a partial algebra $A$ if there is an $S$-algebra whose sorts form a partition of the carrier of algebra $A$ and operations are formed from operations of $A$ by the partition. The main result is that for any partial algebra there is the minimal many sorted signature which characterizes the algebra. The minimality means that there are signature endomorphisms from any signature which characterizes the algebra $A$ onto the minimal one.

- Oleg Okhotnikov.
The Subformula Tree of a Formula of the First Order Language,
Formalized Mathematics 5(3), pages 415-422, 1996. MML Identifier: QC_LANG4

**Summary**: A continuation of \cite{QC_LANG3.ABS}. The notions of list of immediate constituents of a formula and subformula tree of a formula are introduced. The some propositions related to these notions are proved.

- Mariusz Zynel.
The Steinitz Theorem and the Dimension of a Vector Space,
Formalized Mathematics 5(3), pages 423-428, 1996. MML Identifier: VECTSP_9

**Summary**: The main purpose of the paper is to define the dimension of an abstract vector space. The dimension of a finite-dimensional vector space is, by the most common definition, the number of vectors in a basis. Obviously, each basis contains the same number of vectors. We prove the Steinitz Theorem together with Exchange Lemma in the second section. The Steinitz Theorem says that each linearly-independent subset of a vector space has cardinality less than any subset that generates the space, moreover it can be extended to a basis. Further we review some of the standard facts involving the dimension of a vector space. Additionally, in the last section, we introduce two notions: the family of subspaces of a fixed dimension and the pencil of subspaces. Both of them can be applied in the algebraic representation of several geometries.

- Andrzej Trybulec.
On the Go-Board of a Standard Special Circular Sequence,
Formalized Mathematics 5(3), pages 429-438, 1996. MML Identifier: GOBOARD7

**Summary**:

- Jaroslaw Gryko.
On the Monoid of Endomorphisms of Universal Algebra and Many Sorted Algebra,
Formalized Mathematics 5(3), pages 439-442, 1996. MML Identifier: ENDALG

**Summary**:

- Andrzej Trybulec.
More on Segments on a Go-Board,
Formalized Mathematics 5(3), pages 443-450, 1996. MML Identifier: GOBOARD8

**Summary**: We continue the preparatory work for the Jordan Curve Theorem.

- Artur Kornilowicz.
Certain Facts about Families of Subsets of Many Sorted Sets,
Formalized Mathematics 5(3), pages 451-456, 1996. MML Identifier: MSSUBFAM

**Summary**:

- Beata Madras.
On the Concept of the Triangulation,
Formalized Mathematics 5(3), pages 457-462, 1996. MML Identifier: TRIANG_1

**Summary**:

- Yatsuka Nakamura, Piotr Rudnicki.
Vertex Sequences Induced by Chains,
Formalized Mathematics 5(3), pages 297-304, 1996. MML Identifier: GRAPH_2
- Number 4
- Andrzej Trybulec.
Left and Right Component of the Complement of a Special Closed Curve,
Formalized Mathematics 5(4), pages 465-468, 1996. MML Identifier: GOBOARD9

**Summary**: In the article the concept of the left and right component are introduced. These are the auxiliary notions needed in the proof of Jordan Curve Theorem.

- Grzegorz Bancerek.
Reduction Relations,
Formalized Mathematics 5(4), pages 469-478, 1996. MML Identifier: REWRITE1

**Summary**: The goal of the article is to start the formalization of Knuth-Bendix completion method (see \cite{BachmairDershowitz}, \cite{KlopMiddeldorp} or \cite{HofLinCS}; see also \cite{KnuthBendix}, \cite{Huet81}), i.e. to formalize the concept of the completion of a reduction relation. The completion of a reduction relation $R$ is a complete reduction relation equivalent to $R$ such that convertible elements have the same normal forms. The theory formalized in the article includes concepts and facts concerning normal forms, terminating reductions, Church-Rosser property, and equivalence of reduction relations.

- Robert Milewski.
Lattice of Congruences in Many Sorted Algebra,
Formalized Mathematics 5(4), pages 479-483, 1996. MML Identifier: MSUALG_5

**Summary**:

- Grzegorz Bancerek, Andrzej Trybulec.
Miscellaneous Facts about Functions,
Formalized Mathematics 5(4), pages 485-492, 1996. MML Identifier: FUNCT_7

**Summary**:

- Andrzej Trybulec.
Examples of Category Structures,
Formalized Mathematics 5(4), pages 493-500, 1996. MML Identifier: ALTCAT_2

**Summary**: We continue the formalization of the category theory.

- Adam Grabowski.
On the Category of Posets,
Formalized Mathematics 5(4), pages 501-505, 1996. MML Identifier: ORDERS_3

**Summary**: In the paper the construction of a category of partially ordered sets is shown: in the second section according to \cite{CAT_1.ABS} and in the third section according to the definition given in \cite{ALTCAT_1.ABS}. Some of useful notions such as monotone map and the set of monotone maps between relational structures are given.

- Andrzej Trybulec, Yatsuka Nakamura, Piotr Rudnicki.
An Extension of \bf SCM,
Formalized Mathematics 5(4), pages 507-512, 1996. MML Identifier: SCMFSA_1

**Summary**:

- Yatsuka Nakamura, Andrzej Trybulec.
Components and Unions of Components,
Formalized Mathematics 5(4), pages 513-517, 1996. MML Identifier: CONNSP_3

**Summary**: First, we generalized {\bf skl} function for a subset of topological spaces the value of which is the component including the set. Second, we introduced a concept of union of components a family of which has good algebraic properties. At the end, we discuss relationship between connectivity of a set as a subset in the whole space and as a subset of a subspace.

- Andrzej Trybulec, Yatsuka Nakamura, Piotr Rudnicki.
The \SCMFSA Computer,
Formalized Mathematics 5(4), pages 519-528, 1996. MML Identifier: SCMFSA_2

**Summary**:

- Artur Kornilowicz.
On the Many Sorted Closure Operator and the Many Sorted Closure System,
Formalized Mathematics 5(4), pages 529-536, 1996. MML Identifier: CLOSURE1

**Summary**:

- Andrzej Trybulec, Yatsuka Nakamura.
Computation in \SCMFSA,
Formalized Mathematics 5(4), pages 537-542, 1996. MML Identifier: SCMFSA_3

**Summary**: The properties of computations in ${\bf SCM}_{\rm FSA}$ are investigated.

- Artur Kornilowicz.
On the Closure Operator and the Closure System of Many Sorted Sets,
Formalized Mathematics 5(4), pages 543-551, 1996. MML Identifier: CLOSURE2

**Summary**: In this paper definitions of many sorted closure system and many sorted closure operator are introduced. These notations are also introduced in \cite{CLOSURE1.ABS}, but in another meaning. In this article closure system is absolutely multiplicative subset family of many sorted sets and in \cite{CLOSURE1.ABS} is many sorted absolutely multiplicative subset family of many sorted sets. Analogously, closure operator is function between many sorted sets and in \cite{CLOSURE1.ABS} is many sorted function from a many sorted set into a many sorted set.

- Grzegorz Bancerek.
Translations, Endomorphisms, and Stable Equational Theories,
Formalized Mathematics 5(4), pages 553-564, 1996. MML Identifier: MSUALG_6

**Summary**: Equational theories of an algebra, i.e. the equivalence relation closed under translations and endomorphisms, are formalized. The correspondence between equational theories and term rewriting systems is discussed in the paper. We get as the main result that any pair of elements of an algebra belongs to the equational theory generated by a set $A$ of axioms iff the elements are convertible w.r.t. term rewriting reduction determined by $A$.\par The theory is developed according to \cite{WECHLER}.

- Robert Milewski.
More on the Lattice of Many Sorted Equivalence Relations,
Formalized Mathematics 5(4), pages 565-569, 1996. MML Identifier: MSUALG_7

**Summary**:

- Andrzej Trybulec, Yatsuka Nakamura.
Modifying Addresses of Instructions of \SCMFSA,
Formalized Mathematics 5(4), pages 571-576, 1996. MML Identifier: SCMFSA_4

**Summary**:

- Czeslaw Bylinski, Piotr Rudnicki.
The Correspondence Between Monotonic Many Sorted Signatures and Well-Founded Graphs. Part I,
Formalized Mathematics 5(4), pages 577-582, 1996. MML Identifier: MSSCYC_1

**Summary**: We prove a number of auxiliary facts about graphs, mainly about vertex sequences of chains and oriented chains. Then we define a graph to be {\em well-founded} if for each vertex in the graph the length of oriented chains ending at the vertex is bounded. A {\em well-founded} graph does not have directed cycles or infinite descending chains. In the second part of the article we prove some auxiliary facts about free algebras and locally-finite algebras.

- Andrzej Trybulec, Yatsuka Nakamura.
Relocability for \SCMFSA,
Formalized Mathematics 5(4), pages 583-586, 1996. MML Identifier: SCMFSA_5

**Summary**:

- Robert Milewski.
More on the Lattice of Congruences in Many Sorted Algebra,
Formalized Mathematics 5(4), pages 587-590, 1996. MML Identifier: MSUALG_8

**Summary**:

- Czeslaw Bylinski, Piotr Rudnicki.
The Correspondence Between Monotonic Many Sorted Signatures and Well-Founded Graphs. Part II,
Formalized Mathematics 5(4), pages 591-593, 1996. MML Identifier: MSSCYC_2

**Summary**: The graph induced by a many sorted signature is defined as follows: the vertices are the symbols of sorts, and if a sort $s$ is an argument of an operation with result sort $t$, then a directed edge $[s, t]$ is in the graph. The key lemma states relationship between the depth of elements of a free many sorted algebra over a signature and the length of directed chains in the graph induced by the signature. Then we prove that a monotonic many sorted signature (every finitely-generated algebra over it is locally-finite) induces a {\em well-founded} graph. The converse holds with an additional assumption that the signature is finitely operated, i.e. there is only a finite number of operations with the given result sort.

- Andrzej Trybulec.
Functors for Alternative Categories,
Formalized Mathematics 5(4), pages 595-608, 1996. MML Identifier: FUNCTOR0

**Summary**: An attempt to define the concept of a functor covering both cases (covariant and contravariant) resulted in a structure consisting of two fields: the object map and the morphism map, the first one mapping the Cartesian squares of the set of objects rather than the set of objects. We start with an auxiliary notion of {\em bifunction}, i.e. a function mapping the Cartesian square of a set $A$ into the Cartesian square of a set $B$. A {\em bifunction} $f$ is said to be {\em covariant} if there is a function $g$ from $A$ into $B$ that $f$ is the Cartesian square of $g$ and $f$ is {\em contravariant} if there is a function $g$ such that $f(o_1,o_2) = \langle g(o_2),g(o_1) \rangle$. The term {\em transformation}, another auxiliary notion, might be misleading. It is not related to natural transformations. A transformation from a many sorted set $A$ indexed by $I$ into a many sorted set $B$ indexed by $J$ w.r.t. a function $f$ from $I$ into $J$ is a (many sorted) function from $A$ into $B \cdot f$. Eventually, the morphism map of a functor from $C_1$ into $C_2$ is a transformation from the arrows of the category $C_1$ into the composition of the object map of the functor and the arrows of $C_2$.\par Several kinds of functor structures have been defined: one-to-one, faithful, onto, full and id-preserving. We were pressed to split property that the composition be preserved into two: comp-preserving (for covariant functors) and comp-reversing (for contravariant functors). We defined also some operation on functors, e.g. the composition, the inverse functor. In the last section it is defined what is meant that two categories are isomorphic (anti-isomorphic).

- Claus Zinn, Wolfgang Jaksch.
Basic Properties of Functor Structures,
Formalized Mathematics 5(4), pages 609-613, 1996. MML Identifier: FUNCTOR1

**Summary**: This article presents some theorems about functor structures. We start with some basic lemmata concerning the composition of functor structures. Then, two theorems about the restriction operator are formulated. Later we show two theorems concerning the properties 'full' and 'faithful' of functor structures which are equivalent to the 'onto' and 'one-to-one' properties of their morphmaps, respectively. Furthermore, we prove some theorems about the inversion of functor structures.

- Noriko Asamoto.
Some Multi Instructions Defined by Sequence of Instructions of \SCMFSA,
Formalized Mathematics 5(4), pages 615-619, 1996. MML Identifier: SCMFSA_7

**Summary**:

- Mariusz Giero.
More on Products of Many Sorted Algebras,
Formalized Mathematics 5(4), pages 621-626, 1996. MML Identifier: PRALG_3

**Summary**: This article is continuation of an article defining products of many sorted algebras \cite{PRALG_2.ABS}. Some properties of notions such as commute, Frege, Args() are shown in this article. Notions of constant of operations in many sorted algebras and projection of products of family of many sorted algebras are defined. There is also introduced the notion of class of family of many sorted algebras. The main theorem states that product of family of many sorted algebras and product of class of family of many sorted algebras are isomorphic.

- Andrzej Trybulec.
Left and Right Component of the Complement of a Special Closed Curve,
Formalized Mathematics 5(4), pages 465-468, 1996. MML Identifier: GOBOARD9
**Volume 6, 1997**- Number 1
- Yatsuka Nakamura, Andrzej Trybulec.
Adjacency Concept for Pairs of Natural Numbers,
Formalized Mathematics 6(1), pages 1-3, 1997. MML Identifier: GOBRD10

**Summary**: First, we introduce the concept of adjacency for a pair of natural numbers. Second, we extend the concept for two pairs of natural numbers. The pairs represent points of a lattice in a plane. We show that if some property is infectious among adjacent points, and some points have the property, then all points have the property.

- Adam Grabowski.
Inverse Limits of Many Sorted Algebras,
Formalized Mathematics 6(1), pages 5-8, 1997. MML Identifier: MSALIMIT

**Summary**: This article introduces the construction of an inverse limit of many sorted algebras. A few preliminary notions such as an ordered family of many sorted algebras and a binding of family are formulated. Definitions of a set of many sorted signatures and a set of signature morphisms are also given.

- Artur Kornilowicz.
On the Trivial Many Sorted Algebras and Many Sorted Congruences,
Formalized Mathematics 6(1), pages 9-15, 1997. MML Identifier: MSUALG_9

**Summary**: This paper contains properties of many sorted functions between two many sorted sets. Other theorems describe trivial many sorted algebras. In the last section there are theorems about many sorted congruences, which are defined on many sorted algebras. I have also proved facts about natural epimorphism.

- Adam Grabowski.
Examples of Category Structures,
Formalized Mathematics 6(1), pages 17-20, 1997. MML Identifier: MSINST_1

**Summary**: This article contains definitions of two category structures: the category of many sorted signatures and the category of many sorted algebras. Some facts about these structures are proved.

- Andrzej Trybulec, Yatsuka Nakamura, Noriko Asamoto.
On the Compositions of Macro Instructions. Part I,
Formalized Mathematics 6(1), pages 21-27, 1997. MML Identifier: SCMFSA6A

**Summary**:

- Piotr Rudnicki, Andrzej Trybulec.
Memory Handling for \SCMFSA,
Formalized Mathematics 6(1), pages 29-36, 1997. MML Identifier: SF_MASTR

**Summary**: We introduce some terminology for reasoning about memory used in programs in general and in macro instructions (introduced in \cite{SCMFSA6A.ABS}) in particular. The usage of integer locations and of finite sequence locations by a program is treated separately. We define some functors for selecting memory locations needed for local (temporary) variables in macro instructions. Some semantic properties of the introduced notions are given in terms of executions of macro instructions.

- Yatsuka Nakamura, Andrzej Trybulec.
Some Topological Properties of Cells in $\calE^2_\rmT$,
Formalized Mathematics 6(1), pages 37-40, 1997. MML Identifier: GOBRD11

**Summary**: We examine the topological property of cells (rectangles) in a plane. First, some Fraenkel expressions of cells are shown. Second, it is proved that cells are closed. The last theorem asserts that the closure of the interior of a cell is the same as itself.

- Noriko Asamoto, Yatsuka Nakamura, Piotr Rudnicki, Andrzej Trybulec.
On the Composition of Macro Instructions. Part II,
Formalized Mathematics 6(1), pages 41-47, 1997. MML Identifier: SCMFSA6B

**Summary**: We define the semantics of macro instructions (introduced in \cite{SCMFSA6A.ABS}) in terms of executions of ${\bf SCM}_{\rm FSA}$. In a similar way, we define the semantics of macro composition. Several attributes of macro instructions are introduced (paraclosed, parahalting, keeping 0) and their usage enables a systematic treatment of the composition of macro intructions. This article is continued in \cite{SCMFSA6C.ABS}.

- Yatsuka Nakamura, Andrzej Trybulec.
The First Part of Jordan's Theorem for Special Polygons,
Formalized Mathematics 6(1), pages 49-51, 1997. MML Identifier: GOBRD12

**Summary**: We prove here the first part of Jordan's theorem for special polygons, i.e., the complement of a special polygon is the union of two components (a left component and a right component). At this stage, we do not know if the two components are different from each other.

- Noriko Asamoto, Yatsuka Nakamura, Piotr Rudnicki, Andrzej Trybulec.
On the Composition of Macro Instructions. Part III,
Formalized Mathematics 6(1), pages 53-57, 1997. MML Identifier: SCMFSA6C

**Summary**: This article is a continuation of \cite{SCMFSA6A.ABS} and \cite{SCMFSA6C.ABS}. First, we recast the semantics of the macro composition in more convenient terms. Then, we introduce terminology and basic properties of macros constructed out of single instructions of ${\bf SCM}_{\rm FSA}$. We give the complete semantics of composing a macro instruction with an instruction and for composing two machine instructions (this is also done in terms of macros). The introduced terminology is tested on the simple example of a macro for swapping two integer locations.

- Noriko Asamoto.
Constant Assignment Macro Instructions of \SCMFSA. Part II,
Formalized Mathematics 6(1), pages 59-63, 1997. MML Identifier: SCMFSA7B

**Summary**:

- Noriko Asamoto.
Conditional Branch Macro Instructions of \SCMFSA. Part I,
Formalized Mathematics 6(1), pages 65-72, 1997. MML Identifier: SCMFSA8A

**Summary**:

- Noriko Asamoto.
Conditional Branch Macro Instructions of \SCMFSA. Part II,
Formalized Mathematics 6(1), pages 73-80, 1997. MML Identifier: SCMFSA8B

**Summary**:

- Grzegorz Bancerek.
Bounds in Posets and Relational Substructures,
Formalized Mathematics 6(1), pages 81-91, 1997. MML Identifier: YELLOW_0

**Summary**: Notation and facts necessary to start with the formalization of continuous lattices according to \cite{CCL} are introduced.

- Grzegorz Bancerek.
Directed Sets, Nets, Ideals, Filters, and Maps,
Formalized Mathematics 6(1), pages 93-107, 1997. MML Identifier: WAYBEL_0

**Summary**: Notation and facts necessary to start with the formalization of continuous lattices according to \cite{CCL} are introduced. The article contains among other things, the definition of directed and filtered subsets of a poset (see 1.1 in \cite[p.~2]{CCL}), the definition of nets on the poset (see 1.2 in \cite[p.~2]{CCL}), the definition of ideals and filters and the definition of maps preserving arbitrary and directed sups and arbitrary and filtered infs (1.9 also in \cite[p.~4]{CCL}). The concepts of semilattices, sup-semiletices and poset lattices (1.8 in \cite[p.~4]{CCL}) are also introduced. A number of facts concerning the above notion and including remarks 1.4, 1.5, and 1.10 from \cite[pp.~3--5]{CCL} is presented.

- Piotr Rudnicki, Andrzej Trybulec.
Fixpoints in Complete Lattices,
Formalized Mathematics 6(1), pages 109-115, 1997. MML Identifier: KNASTER

**Summary**: Theorem (5) states that if an iterate of a function has a unique fixpoint then it is also the fixpoint of the function. It has been included here in response to P. Andrews claim that such a proof in set theory takes thousands of lines when one starts with the axioms. While probably true, such a claim is misleading about the usefulness of proof-checking systems based on set theory.\par Next, we prove the existence of the least and the greatest fixpoints for $\subseteq$-monotone functions from a powerset to a powerset of a set. Scheme {\it Knaster} is the Knaster theorem about the existence of fixpoints, cf. \cite{lns82}. Theorem (11) is the Banach decomposition theorem which is then used to prove the Schr\"{o}der-Bernstein theorem (12) (we followed Paulson's development of these theorems in Isabelle \cite{lcp95}). It is interesting to note that the last theorem when stated in Mizar in terms of cardinals becomes trivial to prove as in the Mizar development of cardinals the $\leq$ relation is synonymous with $\subseteq$.\par Section 3 introduces the notion of the lattice of a lattice subset provided the subset has lubs and glbs.\par The main theorem of Section 4 is the Tarski theorem (43) that every monotone function $f$ over a complete lattice $L$ has a complete lattice of fixpoints. As the consequence of this theorem we get the existence of the least fixpoint equal to $f^\beta(\bot_L)$ for some ordinal $\beta$ with cardinality not bigger than the cardinality of the carrier of $L$, cf. \cite{lns82}, and analogously the existence of the greatest fixpoint equal to $f^\beta(\top_L)$.\par Section 5 connects the fixpoint properties of monotone functions over complete lattices with the fixpoints of $\subseteq$-monotone functions over the lattice of subsets of a set (Boolean lattice).

- Adam Grabowski, Robert Milewski.
Boolean Posets, Posets under Inclusion and Products of Relational Structures,
Formalized Mathematics 6(1), pages 117-121, 1997. MML Identifier: YELLOW_1

**Summary**: In the paper some notions useful in formalization of \cite{CCL} are introduced, e.g. the definition of the poset of subsets of a set with inclusion as an ordering relation. Using the theory of many sorted sets authors formulate the definition of product of relational structures.

- Mariusz Zynel, Czeslaw Bylinski.
Properties of Relational Structures, Posets, Lattices and Maps,
Formalized Mathematics 6(1), pages 123-130, 1997. MML Identifier: YELLOW_2

**Summary**: In the paper we present some auxiliary facts concerning posets and maps between them. Our main purpose, however is to give an account on complete lattices and lattices of ideals. A sufficient condition that a lattice might be complete, the fixed-point theorem and two remarks upon images of complete lattices in monotone maps, introduced in \cite[pp. 8--9]{CCL}, can be found in Section~7. Section~8 deals with lattices of ideals. We examine the meet and join of two ideals. In order to show that the lattice of ideals is complete, the infinite intersection of ideals is investigated.

- Czeslaw Bylinski.
Galois Connections,
Formalized Mathematics 6(1), pages 131-143, 1997. MML Identifier: WAYBEL_1

**Summary**: The paper is the Mizar encoding of the chapter 0 section 3 of \cite{CCL} In the paper the following concept are defined: Galois connections, Heyting algebras, and Boolean algebras.

- Artur Kornilowicz.
Cartesian Products of Relations and Relational Structures,
Formalized Mathematics 6(1), pages 145-152, 1997. MML Identifier: YELLOW_3

**Summary**: In this paper the definitions of cartesian products of relations and relational structures are introduced. Facts about these notions are proved. This work is the continuation of formalization of \cite{CCL}.

- Artur Kornilowicz.
Definitions and Properties of the Join and Meet of Subsets,
Formalized Mathematics 6(1), pages 153-158, 1997. MML Identifier: YELLOW_4

**Summary**: This paper is the continuation of formalization of \cite{CCL}. The definitions of meet and join of subsets of relational structures are introduced. The properties of these notions are proved.

- Artur Kornilowicz.
Meet--Continuous Lattices,
Formalized Mathematics 6(1), pages 159-167, 1997. MML Identifier: WAYBEL_2

**Summary**: The aim of this work is the formalization of Chapter 0 Section 4 of \cite{CCL}. In this paper the definition of meet-continuous lattices is introduced. Theorem 4.2 and Remark 4.3 are proved.

- Grzegorz Bancerek.
The ``Way-Below'' Relation,
Formalized Mathematics 6(1), pages 169-176, 1997. MML Identifier: WAYBEL_3

**Summary**: In the paper the ``way-below" relation, in symbols $x \ll y$, is introduced. Some authors prefer the term ``relatively compact" or ``way inside", since in the poset of open sets of a topology it is natural to read $U \ll V$ as ``$U$ is relatively compact in $V$". A compact element of a poset (or an element isolated from below) is defined to be way below itself. So, the compactness in the poset of open sets of a topology is equivalent to the compactness in that topology.\par The article includes definitions, facts and examples 1.1--1.8 presented in \cite[pp.~38--42]{CCL}.

- Yatsuka Nakamura, Andrzej Trybulec.
Adjacency Concept for Pairs of Natural Numbers,
Formalized Mathematics 6(1), pages 1-3, 1997. MML Identifier: GOBRD10
- Number 2
- Adam Grabowski.
Auxiliary and Approximating Relations,
Formalized Mathematics 6(2), pages 179-188, 1997. MML Identifier: WAYBEL_4

**Summary**: The aim of this paper is to formalize the second part of Chapter I Section 1 (1.9--1.19) in \cite{CCL}. Definitions of Scott's auxiliary and approximating relations are introduced in this work. We showed that in a meet-continuous lattice, the way-below relation is the intersection of all approximating auxiliary relations (proposition (40) --- compare 1.13 in \cite[pp.~43--47]{CCL}). By (41) a continuous lattice is a complete lattice in which $\ll$ is the smallest approximating auxiliary relation. The notions of the strong interpolation property and the interpolation property are also introduced.

- Katsumi Wasaki, Pauline N. Kawamoto.
2's Complement Circuit,
Formalized Mathematics 6(2), pages 189-197, 1997. MML Identifier: TWOSCOMP

**Summary**: This article introduces various Boolean operators which are used in discussing the properties and stability of a 2's complement circuit. We present the definitions and related theorems for the following logical operators which include negative input/output: 'and2a', 'or2a', 'xor2a' and 'nand2a', 'nor2a', etc. We formalize the concept of a 2's complement circuit, define the structures of complementors/incrementors for binary operations, and prove the stability of the circuit.

- Mariusz Zynel.
The Equational Characterization of Continuous Lattices,
Formalized Mathematics 6(2), pages 199-205, 1997. MML Identifier: WAYBEL_5

**Summary**: The class of continuous lattices can be characterized by infinitary equations. Therefore, it is closed under the formation of subalgebras and homomorphic images. Following the terminology of \cite{Johnstone} we introduce a continuous lattice subframe to be a sublattice closed under the formation of arbitrary infs and directed sups. This notion corresponds with a subalgebra of a continuous lattice in \cite{CCL}.\par The class of completely distributive lattices is also introduced in the paper. Such lattices are complete and satisfy the most restrictive type of the general distributivity law. Obviously each completely distributive lattice is a Heyting algebra. It was hard to find the best Mizar implementation of the complete distributivity equational condition (denoted by CD in \cite{CCL}). The powerful and well developed Many Sorted Theory gives the most convenient way of this formalization. A set double indexed by $K$, introduced in the paper, corresponds with a family $\{x_{j,k}: j\in J, k\in K(j)\}$. It is defined to be a suitable many sorted function. Two special functors: $\rm Sups$ and $\rm Infs$ as counterparts of $\rm Sup$ and $\rm Inf$ respectively, introduced in \cite{YELLOW_2.ABS}, are also defined. Originally the equation in Definition~2.4 of \cite[p.~58]{CCL} looks as follows: $${\textstyle\bigwedge}_{j\in J} {\textstyle\bigvee}_{k\in K(j)} x_{j,k} = {\textstyle\bigvee}_{f\in M} {\textstyle\bigwedge}_{j\in J} x_{j,f(j)},$$ where $M$ is the set of functions defined on $J$ with values $f(j)\in K(j)$.

- Agnieszka Julia Marasik.
Miscellaneous Facts about Relation Structure,
Formalized Mathematics 6(2), pages 207-211, 1997. MML Identifier: YELLOW_5

**Summary**: In the article notation and facts necessary to start with formalization of continuous lattices according to \cite{CCL} are introduced.

- Andrzej Trybulec.
Moore-Smith Convergence,
Formalized Mathematics 6(2), pages 213-225, 1997. MML Identifier: YELLOW_6

**Summary**: The paper introduces the concept of a net (a generalized sequence). The goal is to enable the continuation of the translation of \cite{CCL}.

- Grzegorz Bancerek.
Duality in Relation Structures,
Formalized Mathematics 6(2), pages 227-232, 1997. MML Identifier: YELLOW_7

**Summary**:

- Beata Madras.
Irreducible and Prime Elements,
Formalized Mathematics 6(2), pages 233-239, 1997. MML Identifier: WAYBEL_6

**Summary**: In the paper open and order generating subsets are defined. Irreducible and prime elements are also defined. The article includes definitions and facts presented in \cite[pp.~68--72]{CCL}.

- Grzegorz Bancerek.
Prime Ideals and Filters,
Formalized Mathematics 6(2), pages 241-247, 1997. MML Identifier: WAYBEL_7

**Summary**: The part of \cite[pp.~73--77]{CCL}, i.e. definitions and propositions 3.16--3.27, is formalized in the paper.

- Robert Milewski.
Algebraic Lattices,
Formalized Mathematics 6(2), pages 249-254, 1997. MML Identifier: WAYBEL_8

**Summary**:

- Yatsuka Nakamura, Roman Matuszewski.
Reconstructions of Special Sequences,
Formalized Mathematics 6(2), pages 255-263, 1997. MML Identifier: JORDAN3

**Summary**: We discuss here some methods for reconstructing special sequences which generate special polygonal arcs in ${\cal E}^{2}_{\rm T}$. For such reconstructions we introduce a ``mid" function which cuts out the middle part of a sequence; the ``$\downharpoonleft$" function, which cuts down the left part of a sequence at some point; the ``$\downharpoonright$" function for cutting down the right part at some point; and the ``$\downharpoonleft \downharpoonright$" function for cutting down both sides at two given points.\par We also introduce some methods glueing two special sequences. By such cutting and glueing methods, the speciality of sequences (generatability of special polygonal arcs) is shown to be preserved.

- Adam Naumowicz.
Conjugate Sequences, Bounded Complex Sequences and Convergent Complex Sequences,
Formalized Mathematics 6(2), pages 265-268, 1997. MML Identifier: COMSEQ_2

**Summary**: This article is a continuation of \cite{COMSEQ_1.ABS}. It is divided into five sections. The first one contains a few useful lemmas. In the second part there is a definition of conjugate sequences and proofs of some basic properties of such sequences. The third segment treats of bounded complex sequences,next one contains description of convergent complex sequences. The last and the biggest part of the article contains proofs of main theorems concerning the theory of bounded and convergent complex sequences.

- Artur Kornilowicz.
On the Topological Properties of Meet-Continuous Lattices,
Formalized Mathematics 6(2), pages 269-277, 1997. MML Identifier: WAYBEL_9

**Summary**: This work is continuation of formalization of \cite{CCL}. Proposition 4.4 from Chapter 0 is proved.

- Grzegorz Bancerek.
Institution of Many Sorted Algebras. Part I: Signature Reduct of an Algebra,
Formalized Mathematics 6(2), pages 279-287, 1997. MML Identifier: INSTALG1

**Summary**: In the paper the notation necessary to construct the institution of many sorted algebras is introduced.

- Andrzej Trybulec.
Baire Spaces, Sober Spaces,
Formalized Mathematics 6(2), pages 289-294, 1997. MML Identifier: YELLOW_8

**Summary**: In the article concepts and facts necessary to continue formalization of theory of continuous lattices according to \cite{CCL} are introduced.

- Grzegorz Bancerek.
Closure Operators and Subalgebras,
Formalized Mathematics 6(2), pages 295-301, 1997. MML Identifier: WAYBEL10

**Summary**:

- Grzegorz Bancerek.
Algebra of Morphisms,
Formalized Mathematics 6(2), pages 303-310, 1997. MML Identifier: CATALG_1

**Summary**:

- Andrzej Trybulec.
Scott Topology,
Formalized Mathematics 6(2), pages 311-319, 1997. MML Identifier: WAYBEL11

**Summary**: In the article we continue the formalization in Mizar of \cite[98--105]{CCL}. We work with structures of the form $$L = \langle C,\ \leq,\ \tau \rangle,$$ where $C$ is the carrier of the structure, $\leq$ - an ordering relation on $C$ and $\tau$ a family of subsets of $C$. When $\langle C,\ \leq \rangle$ is a complete lattice we say that $L$ is Scott, if $\tau$ is the Scott topology of $\langle C,\ \leq \rangle$. We define the Scott convergence (lim inf convergence). Following \cite{CCL} we prove that in the case of a continuous lattice $\langle C,\ \leq \rangle$ the Scott convergence is topological, i.e. enjoys the properties: (CONSTANTS), (SUBNETS), (DIVERGENCE), (ITERATED LIMITS). We formalize the theorem, that if the Scott convergence has the (ITERATED LIMITS) property, the $\langle C,\ \leq \rangle$ is continuous.

- Artur Kornilowicz.
On the Baire Category Theorem,
Formalized Mathematics 6(2), pages 321-327, 1997. MML Identifier: WAYBEL12

**Summary**: In this paper Exercise 3.43 from Chapter 1 of \cite{CCL} is solved.

- Adam Grabowski.
Auxiliary and Approximating Relations,
Formalized Mathematics 6(2), pages 179-188, 1997. MML Identifier: WAYBEL_4
- Number 3
- Beata Madras.
Basic Properties of Objects and Morphisms,
Formalized Mathematics 6(3), pages 329-334, 1997. MML Identifier: ALTCAT_3

**Summary**:

- Piotr Rudnicki, Andrzej Trybulec.
Abian's Fixed Point Theorem,
Formalized Mathematics 6(3), pages 335-338, 1997. MML Identifier: ABIAN

**Summary**: A. Abian \cite{abi68} proved the following theorem: \begin{quotation} Let $f$ be a mapping from a finite set $D$. Then $f$ has a fixed point if and only if $D$ is not a union of three mutually disjoint sets $A$, $B$ and $C$ such that \[ A \cap f[A] = B \cap f[B] = C \cap f[C] = \emptyset.\] \end{quotation} (The range of $f$ is not necessarily the subset of its domain). The proof of the sufficiency is by induction on the number of elements of $D$. A.~M\c{a}kowski and K.~Wi{\'s}niewski \cite{maw69} have shown that the assumption of finiteness is superfluous. They proved their version of the theorem for $f$ being a function from $D$ into $D$. In the proof, the required partition was constructed and the construction used the axiom of choice. Their main point was to demonstrate that the use of this axiom in the proof is essential. We have proved in Mizar the generalized version of Abian's theorem, i.e. without assuming finiteness of $D$. We have simplified the proof from \cite{maw69} which uses well-ordering principle and transfinite ordinals---our proof does not use these notions but otherwise is based on their idea (we employ choice functions).

- Piotr Rudnicki, Andrzej Trybulec.
On Same Equivalents of Well-foundedness,
Formalized Mathematics 6(3), pages 339-343, 1997. MML Identifier: WELLFND1

**Summary**: Four statements equivalent to well-foundedness (well-founded induction, existence of recursively defined functions, uniqueness of recursively defined functions, and absence of descending $\omega$-chains) have been proved in Mizar and the proofs were mechanically checked for correctness. It seems not to be widely known that the existence (without the uniqueness assumption) of recursively defined functions implies well-foundedness. In the proof we used regular cardinals, a fairly advanced notion of set theory. This work was inspired by T.~Franzen's paper ~\cite{tor96}. Franzen's proofs were written by a mathematician having an argument with a computer scientist. We were curious about the effort needed to formalize Franzen's proofs given the state of the Mizar Mathematical Library at that time (July 1996). The formalization went quite smoothly once the mathematics was sorted out.

- Robert Milewski.
Algebraic and Arithmetic Lattices,
Formalized Mathematics 6(3), pages 345-349, 1997. MML Identifier: WAYBEL13

**Summary**: We formalize \cite[pp. 87--89]{CCL}.

- Yatsuka Nakamura, Roman Matuszewski, Adam Grabowski.
Subsequences of Standard Special Circular Sequences in $\calE^2_\rmT$,
Formalized Mathematics 6(3), pages 351-358, 1997. MML Identifier: JORDAN4

**Summary**: It is known that a standard special circular sequence in ${\cal E}^2_{\rm T}$ properly defines a special polygon. We are interested in a part of such a sequence. It is shown that if the first point and the last point of the subsequence are different, it becomes a special polygonal sequence. The concept of ``a part of" is introduced, and the subsequence having this property can be characterized by using ``mid" function. For such subsequences, the concepts of ``Upper" and ``Lower" parts are introduced.

- Adam Grabowski.
Lattice of Substitutions,
Formalized Mathematics 6(3), pages 359-361, 1997. MML Identifier: SUBSTLAT

**Summary**:

- Artur Kornilowicz.
Equations in Many Sorted Algebras,
Formalized Mathematics 6(3), pages 363-369, 1997. MML Identifier: EQUATION

**Summary**: This paper is preparation to prove Birkhoff's Theorem. Some properties of many sorted algebras are proved. The last section of this work shows that every equation valid in a many sorted algebra is also valid in each subalgebra, and each image of it. Moreover for a family of many sorted algebras $(A_i: i \in I)$ if every equation is valid in each $A_i$, $i \in I$ then is also valid in product $\prod(A_i: i \in I)$.

- Robert Nieszczerzewski.
Category of Functors Between Alternative Categories,
Formalized Mathematics 6(3), pages 371-375, 1997. MML Identifier: FUNCTOR2

**Summary**:

- Miroslaw Wojciechowski.
Yoneda Embedding,
Formalized Mathematics 6(3), pages 377-379, 1997. MML Identifier: YONEDA_1

**Summary**:

- Christoph Schwarzweller.
The Correctness of the Generic Algorithms of Brown and Henrici Concerning Addition and Multiplication in Fraction Fields,
Formalized Mathematics 6(3), pages 381-388, 1997. MML Identifier: GCD_1

**Summary**: We prove the correctness of the generic algorithms of Brown and Henrici concerning addition and multiplication in fraction fields of gcd-domains. For that we first prove some basic facts about divisibility in integral domains and introduce the concept of amplesets. After that we are able to define gcd-domains and to prove the theorems of Brown and Henrici which are crucial for the correctness of the algorithms. In the last section we define Mizar functions mirroring their input/output behaviour and prove properties of these functions that ensure the correctness of the algorithms.

- Artur Kornilowicz.
Birkhoff Theorem for Many Sorted Algebras,
Formalized Mathematics 6(3), pages 389-395, 1997. MML Identifier: BIRKHOFF

**Summary**: \newcommand \pred[1]{${\cal P} #1$} In this article Birkhoff Variety Theorem for many sorted algebras is proved. A class of algebras is represented by predicate \pred{}. Notation \pred{[A]}, where $A$ is an algebra, means that $A$ is in class \pred{}. All algebras in our class are many sorted over many sorted signature $S$. The properties of varieties: \begin{itemize} \itemsep-3pt \item a class \pred{ } of algebras is abstract \item a class \pred{ } of algebras is closed under subalgebras \item a class \pred{ } of algebras is closed under congruences \item a class \pred{ } of algebras is closed under products \end{itemize} are published in this paper as: \begin{itemize} \itemsep-3pt \item for all non-empty algebras $A$, $B$ over $S$ such that $A$ and $B$ are\_isomorphic and \pred{[A]} holds \pred{[B]} \item for every non-empty algebra $A$ over $S$ and for strict non-empty subalgebra $B$ of $A$ such that \pred{[A]} holds \pred{[B]} \item for every non-empty algebra $A$ over $S$ and for every congruence $R$ of $A$ such that \pred{[A]} holds \pred{[A\slash R]} \item Let $I$ be a set and $F$ be an algebra family of $I$ over ${\cal A}.$ Suppose that for every set $i$ such that $i \in I$ there exists an algebra $A$ over ${\cal A}$ such that $A = F(i)$ and ${\cal P}[A]$. Then${\cal P}[\prod F]$. \end{itemize} This paper is formalization of parts of \cite{WECHLER}.

- Agnieszka Julia Marasik.
Algebraic Operation on Subsets of Many Sorted Sets,
Formalized Mathematics 6(3), pages 397-401, 1997. MML Identifier: CLOSURE3

**Summary**:

- Yasunari Shidama, Artur Kornilowicz.
Convergence and the Limit of Complex Sequences. Series,
Formalized Mathematics 6(3), pages 403-410, 1997. MML Identifier: COMSEQ_3

**Summary**:

- Jing-Chao Chen.
The Steinitz Theorem and the Dimension of a Real Linear Space,
Formalized Mathematics 6(3), pages 411-415, 1997. MML Identifier: RLVECT_5

**Summary**: Finite-dimensional real linear spaces are defined. The dimension of such spaces is the cardinality of a basis. Obviously, each two basis have the same cardinality. We prove the Steinitz theorem and the Exchange Lemma. We also investigate some fundamental facts involving the dimension of real linear spaces.

- Yatsuka Nakamura, Piotr Rudnicki.
Euler Circuits and Paths,
Formalized Mathematics 6(3), pages 417-425, 1997. MML Identifier: GRAPH_3

**Summary**: We prove the Euler theorem on existence of Euler circuits and paths in multigraphs.

- Czeslaw Bylinski, Piotr Rudnicki.
Bounding Boxes for Compact Sets in $\calE^2$,
Formalized Mathematics 6(3), pages 427-440, 1997. MML Identifier: PSCOMP_1

**Summary**: We define pseudocompact topological spaces and prove that every compact space is pseudocompact. We also solve an exercise from \cite{mgm}~p.225 that for a topological space $X$ the following are equivalent: \begin{itemize} \item Every continuous real map from $X$ is bounded (i.e. $X$ is pseudocompact). \item Every continuous real map from $X$ attains minimum. \item Every continuous real map from $X$ attains maximum. \end{itemize} Finally, for a compact set in $E^2$ we define its bounding rectangle and introduce a collection of notions associated with the box.

- Czeslaw Bylinski, Piotr Rudnicki.
The Scott Topology. Part II,
Formalized Mathematics 6(3), pages 441-446, 1997. MML Identifier: WAYBEL14

**Summary**: Mizar formalization of pp. 105--108 of \cite{CCL} which continues \cite{WAYBEL11.ABS}. We found a simplification for the proof of Corollary~1.15, in the last case, see the proof in the Mizar article for details.

- Beata Madras.
Basic Properties of Objects and Morphisms,
Formalized Mathematics 6(3), pages 329-334, 1997. MML Identifier: ALTCAT_3
- Number 4
- Adam Grabowski.
Introduction to the Homotopy Theory,
Formalized Mathematics 6(4), pages 449-454, 1997. MML Identifier: BORSUK_2

**Summary**: The paper introduces some preliminary notions concerning the homotopy theory according to \cite{Greenberg}: paths and arcwise connected to topological spaces. The basic operations on paths (addition and reversing) are defined. In the last section the predicate: $P, Q$ {\em are homotopic} is defined. We also showed some properties of the product of two topological spaces needed to prove reflexivity and symmetry of the above predicate.

- Adam Grabowski, Yatsuka Nakamura.
Some Properties of Real Maps,
Formalized Mathematics 6(4), pages 455-459, 1997. MML Identifier: JORDAN5A

**Summary**: The main goal of the paper is to show logical equivalence of the two definitions of the {\em open subset}: one from \cite{PCOMPS_1.ABS} and the other from \cite{RCOMP_1.ABS}. This has been used to show that the other two definitions are equivalent: the continuity of the map as in \cite{PRE_TOPC.ABS} and in \cite{FCONT_1.ABS}. We used this to show that continuous and one-to-one maps are monotone (see theorems 16 and 17 for details).

- Adam Grabowski, Yatsuka Nakamura.
The Ordering of Points on a Curve. Part I,
Formalized Mathematics 6(4), pages 461-465, 1997. MML Identifier: JORDAN5B

**Summary**: Some auxiliary theorems needed to formalize the proof of the Jordan Curve Theorem according to \cite{TAKE-NAKA} are proved.

- Adam Grabowski, Yatsuka Nakamura.
The Ordering of Points on a Curve. Part II,
Formalized Mathematics 6(4), pages 467-473, 1997. MML Identifier: JORDAN5C

**Summary**: The proof of the Jordan Curve Theorem according to \cite{TAKE-NAKA} is continued. The notions of the first and last point of a oriented arc are introduced as well as ordering of points on a curve in $\calE^2_T$.

- Artur Kornilowicz.
On the Categories Without Uniqueness of \bf cod and \bf dom . Some Properties of the Morphisms and the Functors,
Formalized Mathematics 6(4), pages 475-481, 1997. MML Identifier: ALTCAT_4

**Summary**:

- Noriko Asamoto.
The loop and Times Macroinstruction for \SCMFSA,
Formalized Mathematics 6(4), pages 483-497, 1997. MML Identifier: SCMFSA8C

**Summary**: We implement two macroinstructions {\tt loop} and {\tt Times} which iterate macroinstructions of {\SCMFSA}. In a {\tt loop} macroinstruction it jumps to the head when the original macroinstruction stops, in a {\tt Times} macroinstruction it behaves as if the original macroinstruction repeats $n$ times.

- Robert Milewski.
Algebraic and Arithmetic Lattices. Part II,
Formalized Mathematics 6(4), pages 499-503, 1997. MML Identifier: WAYBEL15

**Summary**: The article is a translation of \cite[pp. 89--92]{CCL}.

- Roman Matuszewski, Yatsuka Nakamura.
Projections in n-Dimensional Euclidean Space to Each Coordinates,
Formalized Mathematics 6(4), pages 505-509, 1997. MML Identifier: JORDAN2B

**Summary**: In the $n$-dimensional Euclidean space ${\cal E}^n_{\rm T}$, a projection operator to each coordinate is defined. It is proven that such an operator is linear. Moreover, it is continuous as a mapping from ${\cal E}^n_{\rm T}$ to ${R}^{1}$, the carrier of which is a set of all reals. If $n$ is 1, the projection becomes a homeomorphism, which means that ${\cal E}^1_{\rm T}$ is homeomorphic to ${R}^{1}$.

- Yatsuka Nakamura, Andrzej Trybulec.
Intermediate Value Theorem and Thickness of Simple Closed Curves,
Formalized Mathematics 6(4), pages 511-514, 1997. MML Identifier: TOPREAL5

**Summary**: Various types of the intermediate value theorem (\cite {shilov}) are proved. For their special cases, the Bolzano theorem is also proved. Using such a theorem, it is shown that if a curve is a simple closed curve, then it is not horizontally degenerated, neither is it vertically degenerated.

- Jaroslaw Gryko.
The J\'onson's Theorem,
Formalized Mathematics 6(4), pages 515-524, 1997. MML Identifier: LATTICE5

**Summary**:

- Yatsuka Nakamura, Andrzej Trybulec.
Lebesgue's Covering Lemma, Uniform Continuity and Segmentation of Arcs,
Formalized Mathematics 6(4), pages 525-529, 1997. MML Identifier: UNIFORM1

**Summary**: For mappings from a metric space to a metric space, a notion of uniform continuity is defined. If we introduce natural topologies to the metric spaces, a uniformly continuous function becomes continuous. On the other hand, if the domain is compact, a continuous function is uniformly continuous. For this proof, Lebesgue's covering lemma is also proved. An arc, which is homeomorphic to [0,1], can be divided into small segments, as small as one wishes.

- Andrzej Trybulec, Yatsuka Nakamura.
On the Rectangular Finite Sequences of the Points of the Plane,
Formalized Mathematics 6(4), pages 531-539, 1997. MML Identifier: SPRECT_1

**Summary**: The article deals with a rather technical concept -- rectangular sequences of the points of the plane. We mean by that a finite sequence consisting of five elements, that is circular, i.e. the first element and the fifth one of it are equal, and such that the polygon determined by it is a non degenerated rectangle, with sides parallel to axes. The main result is that for the rectangle determined by such a sequence the left and the right component of the complement of it are different and disjoint.

- Andrzej Trybulec, Yatsuka Nakamura.
On the Order on a Special Polygon,
Formalized Mathematics 6(4), pages 541-548, 1997. MML Identifier: SPRECT_2

**Summary**: The goal of the article is to determine the order of the special points defined in \cite{PSCOMP_1.ABS} on a special polygon. We restrict ourselves to the clockwise oriented finite sequences (the concept defined in this article) that start in N-min C (C being a compact non empty subset of the plane).

- Yoshinori Fujisawa, Yasushi Fuwa.
The Euler's Function,
Formalized Mathematics 6(4), pages 549-551, 1997. MML Identifier: EULER_1

**Summary**: This article is concerned with the Euler's function \cite{takagi} that plays an important role in cryptograms. In the first section, we present some selected theorems on integers. Next, we define the Euler's function. Finally, three theorems relating to the Euler's function are proved. The third theorem concerns two relatively prime integers which make up the Euler's function parameter. In the public key cryptography these two integer values are used as public and secret keys.

- Jing-Chao Chen.
While Macro Instructions of \SCMFSA,
Formalized Mathematics 6(4), pages 553-561, 1997. MML Identifier: SCMFSA_9

**Summary**: The article defines {\em while macro instructions} based on \SCMFSA. Some theorems about the generalized halting problems of {\em while macro instructions} are proved.

- Yatsuka Nakamura, Andrzej Trybulec.
A Decomposition of a Simple Closed Curves and the Order of Their Points,
Formalized Mathematics 6(4), pages 563-572, 1997. MML Identifier: JORDAN6

**Summary**: The goal of the article is to introduce an order on a simple closed curve. To do this, we fix two points on the curve and devide it into two arcs. We prove that such a decomposition is unique. Other auxiliary theorems about arcs are proven for preparation of the proof of the above.

- Andrzej Kondracki.
The Chinese Remainder Theorem,
Formalized Mathematics 6(4), pages 573-577, 1997. MML Identifier: WSIERP_1

**Summary**: The article is a translation of the first chapters of a book {\em Wst{\Ple}p do teorii liczb} (Eng. {\em Introduction to Number Theory}) by W. Sierpi\'nski, WSiP, Biblioteczka Matematyczna, Warszawa, 1987. The first few pages of this book have already been formalized in MML. We prove the Chinese Remainder Theorem and Thue's Theorem as well as several useful number theory propositions.

- Adam Grabowski.
Introduction to the Homotopy Theory,
Formalized Mathematics 6(4), pages 449-454, 1997. MML Identifier: BORSUK_2
**Volume 7, 1998**- Number 1
- Artur Kornilowicz.
The Composition of Functors and Transformations in Alternative Categories,
Formalized Mathematics 7(1), pages 1-7, 1998. MML Identifier: FUNCTOR3

**Summary**:

- Robert Milewski.
Completely-Irreducible Elements,
Formalized Mathematics 7(1), pages 9-12, 1998. MML Identifier: WAYBEL16

**Summary**: The article is a translation of \cite[92--93]{CCL}.

- Adam Grabowski.
Scott-Continuous Functions,
Formalized Mathematics 7(1), pages 13-18, 1998. MML Identifier: WAYBEL17

**Summary**: The article is a translation of \cite[pp. 112--113]{CCL}.

- Robert Milewski.
Natural Numbers,
Formalized Mathematics 7(1), pages 19-22, 1998. MML Identifier: NAT_2

**Summary**:

- Robert Milewski.
Binary Arithmetics. Binary Sequences,
Formalized Mathematics 7(1), pages 23-26, 1998. MML Identifier: BINARI_3

**Summary**:

- Robert Milewski.
Full Trees,
Formalized Mathematics 7(1), pages 27-30, 1998. MML Identifier: BINTREE2

**Summary**:

- Adam Naumowicz, Mariusz Lapinski.
On \Tone\ Reflex of Topological Space,
Formalized Mathematics 7(1), pages 31-34, 1998. MML Identifier: T_1TOPSP

**Summary**: This article contains a definition of $T_{1}$ reflex of a topological space as a quotient space which is $T_{1}$ and fulfils the condition that every continuous map $f$ from a topological space $T$ into $S$ being $T_{1}$ space can be considered as a superposition of two continuous maps: the first from $T$ onto its $T_{1}$ reflex and the last from $T_{1}$ reflex of $T$ into $S$.

- Grzegorz Bancerek.
Bases and Refinements of Topologies,
Formalized Mathematics 7(1), pages 35-43, 1998. MML Identifier: YELLOW_9

**Summary**:

- Artur Kornilowicz.
The Properties of Product of Relational Structures,
Formalized Mathematics 7(1), pages 45-52, 1998. MML Identifier: YELLOW10

**Summary**: This work contains useful facts about the product of relational structures. It continues the formalization of \cite{CCL}.

- Adam Naumowicz.
On the Characterization of Modular and Distributive Lattices,
Formalized Mathematics 7(1), pages 53-55, 1998. MML Identifier: YELLOW11

**Summary**: This article contains definitions of the ``pentagon'' lattice $N_5$ and the ``diamond'' lattice $M_3$. It is followed by the characterization of modular and distributive lattices depending on the possible shape of substructures. The last part treats of interval-like sublattices of any lattice.

- Jaroslaw Gryko.
Injective Spaces,
Formalized Mathematics 7(1), pages 57-62, 1998. MML Identifier: WAYBEL18

**Summary**:

- Artur Kornilowicz.
On the Characterization of Hausdorff Spaces,
Formalized Mathematics 7(1), pages 63-68, 1998. MML Identifier: YELLOW12

**Summary**:

- Christoph Schwarzweller.
The Field of Quotients Over an Integral Domain,
Formalized Mathematics 7(1), pages 69-79, 1998. MML Identifier: QUOFIELD

**Summary**: We introduce the field of quotients over an integral domain following the well-known construction using pairs over integral domains. In addition we define ring homomorphisms and prove some basic facts about fields of quotients including their universal property.

- Bartlomiej Skorulski.
First-countable, Sequential, and Frechet Spaces,
Formalized Mathematics 7(1), pages 81-86, 1998. MML Identifier: FRECHET

**Summary**: This article contains a definition of three classes of topological spaces: first-countable, Frechet, and sequential. Next there are some facts about them, that every first-countable space is Frechet and every Frechet space is sequential. Next section contains a formalized construction of topological space which is Frechet but not first-countable. This article is based on \cite[pp. 73--81]{ENGEL:1}.

- Piotr Rudnicki.
On the Composition of Non-parahalting Macro Instructions,
Formalized Mathematics 7(1), pages 87-92, 1998. MML Identifier: SFMASTR1

**Summary**: An attempt to use the {\tt Times} macro, \cite{SCMFSA8C.ABS}, was the origin of writing this article. First, the semantics of the macro composition as developed in \cite{SCMFSA6A.ABS}, \cite{SCMFSA6B.ABS}, \cite{SCMFSA6C.ABS} is extended to the case of macro instructions which are not always halting. Next, several functors extending the memory handling for {\SCMFSA}, \cite{SF_MASTR.ABS}, are defined; they are convenient when writing more complicated programs. After this preparatory work, we define a macro instruction computing the Fibonacci sequence (see the SCM program computing the same sequence in \cite{FIB_FUSC.ABS}) and prove its correctness. The semantics of the {\tt Times} macro is given in \cite{SCMFSA8C.ABS} only for the case when the iterated instruction is parahalting; this is remedied in \cite{SFMASTR2.ABS}.

- Piotr Rudnicki.
The while Macro Instructions of \SCMFSA. Part II,
Formalized Mathematics 7(1), pages 93-100, 1998. MML Identifier: SCMFSA9A

**Summary**: An attempt to use the {\tt while} macro, \cite{SCMFSA_9.ABS}, was the origin of writing this article. The {\tt while} semantics, as given by J.-C.~Chen, is slightly extended by weakening its correctness conditions and this forced a quite straightforward remake of a number of theorems from \cite{SCMFSA_9.ABS}. Numerous additional properties of the {\tt while} macro are then proven. In the last section, we define a macro instruction computing the {\tt fusc} function (see the SCM program computing the same function in \cite{FIB_FUSC.ABS}) and prove its correctness.

- Piotr Rudnicki.
Another times Macro Instruction,
Formalized Mathematics 7(1), pages 101-105, 1998. MML Identifier: SFMASTR2

**Summary**: The semantics of the {\tt times} macro is given in \cite{SCMFSA8C.ABS} only for the case when the body of the macro is parahalting. We remedy this by defining a new {\tt times} macro instruction in terms of {\tt while} (see \cite{SCMFSA_9.ABS}, \cite{SCMFSA9A.ABS}). The semantics of the new {\tt times} macro is given in a way analogous to the semantics of {\tt while} macros. The new {\tt times} uses an anonymous variable to control the number of its executions. We present two examples: a trivial one and a remake of the macro for the Fibonacci sequence (see \cite{SFMASTR1.ABS}).

- Piotr Rudnicki.
The for (going up) Macro Instruction,
Formalized Mathematics 7(1), pages 107-114, 1998. MML Identifier: SFMASTR3

**Summary**: We define a {\tt for} type (going up) macro instruction in terms of the {\tt while} macro. This gives an iterative macro with an explicit control variable. The {\tt for} macro is used to define a macro for the selection sort acting on a finite sequence location of {\SCMFSA}. On the way, a macro for finding a minimum in a section of an array is defined.

- Yatsuka Nakamura, Adam Grabowski.
Bounding Boxes for Special Sequences in $\calE^2$,
Formalized Mathematics 7(1), pages 115-121, 1998. MML Identifier: JORDAN5D

**Summary**: This is the continuation of the proof of the Jordan Theorem according to \cite{TAKE-NAKA}.

- Yoshinori Fujisawa, Yasushi Fuwa, Hidetaka Shimizu.
Euler's Theorem and Small Fermat's Theorem,
Formalized Mathematics 7(1), pages 123-126, 1998. MML Identifier: EULER_2

**Summary**: This article is concerned with Euler's theorem and small Fermat's theorem that play important roles in public-key cryptograms. In the first section, we present some selected theorems on integers. In the following section, we remake definitions about the finite sequence of natural, the function of natural times finite sequence of natural and $\pi$ of the finite sequence of natural. We also prove some basic theorems that concern these redefinitions. Next, we define the function of modulus for finite sequence of natural and some fundamental theorems about this function are proved. Finally, Euler's theorem and small Fermat's theorem are proved.

- Artur Kornilowicz.
The Product of the Families of the Groups,
Formalized Mathematics 7(1), pages 127-134, 1998. MML Identifier: GROUP_7

**Summary**:

- Yatsuka Nakamura.
On the Dividing Function of the Simple Closed Curve into Segments,
Formalized Mathematics 7(1), pages 135-138, 1998. MML Identifier: JORDAN7

**Summary**: At the beginning, the concept of the segment of the simple closed curve in 2-dimensional Euclidean space is defined. Some properties of segments are shown in the succeeding theorems. At the end, the existence of the function which can divide the simple closed curve into segments is shown. We can make the diameter of segments as small as we want.

- Jing-Chao Chen, Yatsuka Nakamura.
Initialization Halting Concepts and Their Basic Properties of \SCMFSA,
Formalized Mathematics 7(1), pages 139-151, 1998. MML Identifier: SCM_HALT

**Summary**: Up to now, many properties of macro instructions of {\SCMFSA} are described by the parahalting concepts. However, many practical programs are not always halting while they are halting for initialization states. For this reason, we propose initialization halting concepts. That a program is initialization halting (called ``InitHalting'' for short) means it is halting for initialization states. In order to make the halting proof of more complicated programs easy, we present ``InitHalting'' basic properties of the compositions of the macro instructions, if-Macro (conditional branch macro instructions) and Times-Macro (for-loop macro instructions) etc.

- Jing-Chao Chen, Yatsuka Nakamura.
Bubble Sort on \SCMFSA,
Formalized Mathematics 7(1), pages 153-161, 1998. MML Identifier: SCMBSORT

**Summary**: We present the bubble sorting algorithm using macro instructions such as the if-Macro (conditional branch macro instructions) and the Times-Macro (for-loop macro instructions) etc. The correctness proof of the program should include the proof of autonomic, halting and the correctness of the program result. In the three terms, we justify rigorously the correctness of the bubble sorting algorithm. In order to prove it is autonomic, we use the following theorem: if all variables used by the program are initialized, it is autonomic. This justification method probably reveals that autonomic concept is not important.

- Artur Kornilowicz.
The Composition of Functors and Transformations in Alternative Categories,
Formalized Mathematics 7(1), pages 1-7, 1998. MML Identifier: FUNCTOR3
- Number 2
- Grzegorz Bancerek.
The Lawson Topology,
Formalized Mathematics 7(2), pages 163-168, 1998. MML Identifier: WAYBEL19

**Summary**: The article includes definitions, lemmas and theorems 1.1--1.7, 1.9, 1.10 presented in Chapter III of \cite[pp.~142--146]{CCL}.

- Piotr Rudnicki.
Kernel Projections and Quotient Lattices,
Formalized Mathematics 7(2), pages 169-175, 1998. MML Identifier: WAYBEL20

**Summary**: This article completes the Mizar formalization of Chapter I, Section 2 from \cite{CCL}. After presenting some preliminary material (not all of which is later used in this article) we give the proof of theorem 2.7 (i), p.60. We do not follow the hint from \cite{CCL} suggesting using the equations 2.3, p. 58. The proof is taken directly from the definition of continuous lattice. The goal of the last section is to prove the correspondence between the set of all congruences of a continuous lattice and the set of all kernel operators of the lattice which preserve directed sups (Corollary 2.13).

- Grzegorz Bancerek.
Lawson Topology in Continuous Lattices,
Formalized Mathematics 7(2), pages 177-184, 1998. MML Identifier: WAYBEL21

**Summary**: The article completes Mizar formalization of Section 1 of Chapter III of \cite[pp.~145--147]{CCL}.

- Piotr Rudnicki.
Representation Theorem for Free Continuous Lattices,
Formalized Mathematics 7(2), pages 185-188, 1998. MML Identifier: WAYBEL22

**Summary**: We present the Mizar formalization of theorem 4.17, Chapter I from \cite{CCL}: a free continuous lattice with $m$ generators is isomorphic to the lattice of filters of $2^X$ ($\overline{\overline{X}} = m$) which is freely generated by $\{\uparrow x : x \in X\}$ (the set of ultrafilters).

- Yatsuka Nakamura, Piotr Rudnicki.
Oriented Chains,
Formalized Mathematics 7(2), pages 189-192, 1998. MML Identifier: GRAPH_4

**Summary**: In \cite{GRAPH_2.ABS} we introduced a number of notions about vertex sequences associated with undirected chains of edges in graphs. In this article, we introduce analogous concepts for oriented chains and use them to prove properties of cutting and glueing of oriented chains, and the existence of a simple oriented chain in an oriented chain.

- Yatsuka Nakamura.
Graph Theoretical Properties of Arcs in the Plane and Fashoda Meet Theorem,
Formalized Mathematics 7(2), pages 193-201, 1998. MML Identifier: JGRAPH_1

**Summary**: We define a graph on an abstract set, edges of which are pairs of any two elements. For any finite sequence of a plane, we give a definition of nodic, which means that edges by a finite sequence are crossed only at terminals. If the first point and the last point of a finite sequence differs, simpleness as a chain and nodic condition imply unfoldedness and s.n.c. condition. We generalize Goboard Theorem, proved by us before, to a continuous case. We call this Fashoda Meet Theorem, which was taken from Fashoda incident of 100 years ago.

- Yasushi Fuwa, Yoshinori Fujisawa.
Algebraic Group on Fixed-length Bit Integer and its Adaptation to IDEA Cryptography,
Formalized Mathematics 7(2), pages 203-215, 1998. MML Identifier: IDEA_1

**Summary**: In this article, an algebraic group on fixed-length bit integer is constructed and its adaptation to IDEA cryptography is discussed. In the first section, we present some selected theorems on integers. In the continuous section, we construct an algebraic group on fixed-length integer. In the third section, operations of IDEA Cryptograms are defined and some theorems on these operations are proved. In the fourth section, we define sequences of IDEA Cryptogram's operations and discuss their nature. Finally, we make a model of IDEA Cryptogram and prove that the ciphertext that is encrypted by IDEA encryption algorithm can be decrypted by the IDEA decryption algorithm.

- Artur Kornilowicz.
The Definition and Basic Properties of Topological Groups,
Formalized Mathematics 7(2), pages 217-225, 1998. MML Identifier: TOPGRP_1

**Summary**:

- Adam Naumowicz, Agnieszka Julia Marasik.
The Correspondence Between Lattices of Subalgebras of Universal Algebras and Many Sorted Algebras,
Formalized Mathematics 7(2), pages 227-231, 1998. MML Identifier: MSSUBLAT

**Summary**: The main goal of this paper is to show some properties of subalgebras of universal algebras and many sorted algebras, and then the isomorphic correspondence between lattices of such subalgebras.

- Christoph Schwarzweller.
Introduction to Concept Lattices,
Formalized Mathematics 7(2), pages 233-241, 1998. MML Identifier: CONLAT_1

**Summary**: In this paper we give Mizar formalization of concept lattices. Concept lattices stem from the so-called formal concept analysis --- a part of applied mathematics that brings mathematical methods into the field of data analysis and knowledge processing. Our approach follows the one given in \cite{GanterWille}.

- Shunichi Kobayashi, Kui Jia.
A Theory of Partitions. Part I,
Formalized Mathematics 7(2), pages 243-247, 1998. MML Identifier: PARTIT1

**Summary**: In this paper, we define join and meet operations between partitions. The properties of these operations are proved. Then we introduce the correspondence between partitions and equivalence relations which preserve join and meet operations. The properties of these relationships are proved.

- Shunichi Kobayashi, Kui Jia.
A Theory of Boolean Valued Functions and Partitions,
Formalized Mathematics 7(2), pages 249-254, 1998. MML Identifier: BVFUNC_1

**Summary**: In this paper, we define Boolean valued functions. Some of their algebraic properties are proved. We also introduce and examine the infimum and supremum of Boolean valued functions and their properties. In the last section, relations between Boolean valued functions and partitions are discussed.

- Yuguang Yang, Yasunari Shidama.
Trigonometric Functions and Existence of Circle Ratio,
Formalized Mathematics 7(2), pages 255-263, 1998. MML Identifier: SIN_COS

**Summary**: In this article, we defined {\em sinus} and {\em cosine} as the real part and the imaginary part of the exponential function on complex, and also give their series expression. Then we proved the differentiablity of {\em sinus}, {\em cosine} and the exponential function of real. Finally, we showed the existence of the circle ratio, and some formulas of {\em sinus}, {\em cosine}.

- Andrzej Trybulec, Yatsuka Nakamura.
Some Properties of Special Polygonal Curves,
Formalized Mathematics 7(2), pages 265-272, 1998. MML Identifier: SPRECT_3

**Summary**: In the paper some auxiliary theorems are proved, needed in the proof of the second part of the Jordan curve theorem for special polygons. They deal mostly with characteristic points of plane non empty compacts introduced in \cite{PSCOMP_1.ABS}, operation {\em mid} introduced in \cite{JORDAN3.ABS} and the predicate ``$f$ is in the area of $g$'' ($f$ and $g$ : finite sequences of points of the plane) introduced in \cite{SPRECT_2.ABS}.

- Robert Milewski.
Real Linear-Metric Space and Isometric Functions,
Formalized Mathematics 7(2), pages 273-277, 1998. MML Identifier: VECTMETR

**Summary**:

- Artur Kornilowicz.
Introduction to Meet-Continuous Topological Lattices,
Formalized Mathematics 7(2), pages 279-283, 1998. MML Identifier: YELLOW13

**Summary**:

- Robert Milewski.
Bases of Continuous Lattices,
Formalized Mathematics 7(2), pages 285-294, 1998. MML Identifier: WAYBEL23

**Summary**: The article is a Mizar formalization of \cite[168--169]{CCL}. We show definition and fundamental theorems from theory of basis of continuous lattices.

- Artur Kornilowicz.
The Construction of \SCM over Ring,
Formalized Mathematics 7(2), pages 295-300, 1998. MML Identifier: SCMRING1

**Summary**:

- Artur Kornilowicz.
The Basic Properties of \SCM over Ring,
Formalized Mathematics 7(2), pages 301-305, 1998. MML Identifier: SCMRING2

**Summary**:

- Shunichi Kobayashi, Yatsuka Nakamura.
A Theory of Boolean Valued Functions and Quantifiers with Respect to Partitions,
Formalized Mathematics 7(2), pages 307-312, 1998. MML Identifier: BVFUNC_2

**Summary**: In this paper, we define the coordinate of partitions. We also introduce the universal quantifier and the existential quantifier of Boolean valued functions with respect to partitions. Some predicate calculus formulae containing such quantifiers are proved. Such a theory gives a discussion of semantics to usual predicate logic.

- Shunichi Kobayashi, Yatsuka Nakamura.
Predicate Calculus for Boolean Valued Functions. Part I,
Formalized Mathematics 7(2), pages 313-315, 1998. MML Identifier: BVFUNC_3

**Summary**: In this paper, we have proved some elementary predicate calculus formulae containing the quantifiers of Boolean valued functions with respect to partitions. Such a theory is an analogy of usual predicate logic.

- Yoshinori Fujisawa, Yasushi Fuwa, Hidetaka Shimizu.
Public-Key Cryptography and Pepin's Test for the Primality of Fermat Numbers,
Formalized Mathematics 7(2), pages 317-321, 1998. MML Identifier: PEPIN

**Summary**: In this article, we have proved the correctness of the Public-Key Cryptography and the Pepin's Test for the Primality of Fermat Numbers ($F(n) = 2^{2^n}+1$). It is a very important result in the IDEA Cryptography that F(4) is a prime number. At first, we prepared some useful theorems. Then, we proved the correctness of the Public-Key Cryptography. Next, we defined the Order's function and proved some properties. This function is very important in the proof of the Pepin's Test. Next, we proved some theorems about the Fermat Number. And finally, we proved the Pepin's Test using some properties of the Order's Function. And using the obtained result we have proved that F(1), F(2), F(3) and F(4) are prime number.

- Adam Grabowski.
Lattice of Substitutions is a Heyting Algebra,
Formalized Mathematics 7(2), pages 323-327, 1998. MML Identifier: HEYTING2

**Summary**:

- Grzegorz Bancerek.
The Lawson Topology,
Formalized Mathematics 7(2), pages 163-168, 1998. MML Identifier: WAYBEL19
**Volume 8, 1999**- Number 1
- Yatsuka Nakamura, Andrzej Trybulec, Czeslaw Bylinski.
Bounded Domains and Unbounded Domains,
Formalized Mathematics 8(1), pages 1-13, 1999. MML Identifier: JORDAN2C

**Summary**: First, notions of inside components and outside components are introduced for any subset of $n$-dimensional Euclid space. Next, notions of the bounded domain and the unbounded domain are defined using the above components. If the dimension is larger than 1, and if a subset is bounded, a unbounded domain of the subset coincides with an outside component (which is unique) of the subset. For a sphere in $n$-dimensional space, the similar fact is true for a bounded domain. In 2 dimensional space, any rectangle also has such property. We discussed relations between the Jordan property and the concept of boundary, which are necessary to find points in domains near a curve. In the last part, we gave the sufficient criterion for belonging to the left component of some clockwise oriented finite sequences.

- Andrzej Trybulec.
Rotating and Reversing,
Formalized Mathematics 8(1), pages 15-20, 1999. MML Identifier: REVROT_1

**Summary**: Quite a number of lemmas for the Jordan curve theorem, as yet in the case of the special polygonal curves, have been proved. By ``special" we mean, that it is a polygonal curve with edges parallel to axes and actually the lemmas have been proved, mostly, for the triangulations i.e. for finite sequences that define the curve. Moreover some of the results deal only with a special case: \begin{itemize} \item[-] finite sequences are clockwise oriented, \item[-] the first member of the sequence is the member with the lowest ordinate among those with the highest abscissa (N-min $f,$ where $f$ is a finite sequence, in the Mizar jargon). \end{itemize} In the change of the orientation one has to reverse the sequence (the operation introduced in \cite{FINSEQ_5.ABS}) and to change the second restriction one has to rotate the sequence (the operation introduced in \cite{FINSEQ_6.ABS}). The goal of the paper is to prove, mostly simple, facts about the relationship between properties and attributes of the finite sequence and its rotation (similar results about reversing had been proved in \cite{FINSEQ_5.ABS}). Some of them deal with recounting parameters, others with properties that are invariant under the rotation. We prove also that the finite sequence is either clockwise oriented or it is such after reversing. Everything is proved for the so called standard finite sequences, which means that if a point belongs to it then every point with the same abscissa or with the same ordinate, that belongs to the polygon, belongs also to the finite sequence. It does not seem that this requirement causes serious technical obstacles.

- Andrzej Trybulec, Yatsuka Nakamura.
On the Components of the Complement of a Special Polygonal Curve,
Formalized Mathematics 8(1), pages 21-23, 1999. MML Identifier: SPRECT_4

**Summary**: By the special polygonal curve we meana simple closed curve, that is a polygone and moreover has edges parallel to axes. We continue the formalization of the Takeuti-Nakamura proof \cite{TAKE-NAKA} of the Jordan curve theorem. In the paper we prove that the complement of the special polygonal curve consists of at least two components. With the theorem which has at most two components we completed the theorem that a special polygonal curve cuts the plane into exactly two components.

- Czeslaw Bylinski.
Gauges,
Formalized Mathematics 8(1), pages 25-27, 1999. MML Identifier: JORDAN8

**Summary**:

- Christoph Schwarzweller.
The Ring of Integers, Euclidean Rings and Modulo Integers,
Formalized Mathematics 8(1), pages 29-34, 1999. MML Identifier: INT_3

**Summary**: In this article we introduce the ring of Integers, Euclidean rings and Integers modulo $p$. In particular we prove that the Ring of Integers is an Euclidean ring and that the Integers modulo $p$ constitutes a field if and only if $p$ is a prime.

- Yatsuka Nakamura.
Logic Gates and Logical Equivalence of Adders,
Formalized Mathematics 8(1), pages 35-45, 1999. MML Identifier: GATE_1

**Summary**: This is an experimental article which shows that logical correctness of logic circuits can be easily proven by the Mizar system. First, we define the notion of logic gates. Then we prove that an MSB carry of `4 Bit Carry Skip Adder' is equivalent to an MSB carry of a normal 4 bit adder. In the last theorem, we show that outputs of the `4 Bit Carry Look Ahead Adder' are equivalent to the corresponding outputs of the normal 4 bits adder. The policy here is as follows: when the functional (semantic) correctness of a system is already proven, and the correspondence of the system to a (normal) logic circuit is given, it is enough to prove the correctness of the new circuit if we only prove the logical equivalence between them. Although the article is very fundamental (it contains few environment files), it can be applied to real problems. The key of the method introduced here is to put the specification of the logic circuit into the Mizar propositional formulae, and to use the strong inference ability of the Mizar checker. The proof is done formally so that the automation of the proof writing is possible. Even in the 5.3.07 version of Mizar, it can handle a formulae of more than 100 lines, and a formula which contains more than 100 variables. This means that the Mizar system is enough to prove logical correctness of middle scaled logic circuits.

- Bartlomiej Skorulski.
The Sequential Closure Operator in Sequential and Frechet Spaces,
Formalized Mathematics 8(1), pages 47-54, 1999. MML Identifier: FRECHET2

**Summary**:

- Adam Grabowski.
Properties of the Product of Compact Topological Spaces,
Formalized Mathematics 8(1), pages 55-59, 1999. MML Identifier: BORSUK_3

**Summary**:

- Artur Kornilowicz.
Compactness of the Bounded Closed Subsets of $\calE^2_\rmT$,
Formalized Mathematics 8(1), pages 61-68, 1999. MML Identifier: TOPREAL6

**Summary**: This paper contains theorems which describe the correspondence between topological properties of real numbers subsets introduced in \cite{RCOMP_1.ABS} and introduced in \cite{PRE_TOPC.ABS}, \cite{COMPTS_1.ABS}. We also show the homeomorphism between the cartesian product of two $R^1$ and ${\cal E}^2_{\rm T}$. The compactness of the bounded closed subset of ${\cal E}^2_{\rm T}$ is proven.

- Adam Grabowski.
Hilbert Positive Propositional Calculus,
Formalized Mathematics 8(1), pages 69-72, 1999. MML Identifier: HILBERT1

**Summary**:

- Artur Kornilowicz.
Homeomorphism between [:$\calE^i_\rmT, \calE^j_\rmT$:] and $\calE^i+j_\rmT$,
Formalized Mathematics 8(1), pages 73-76, 1999. MML Identifier: TOPREAL7

**Summary**: In this paper we introduce the cartesian product of two metric spaces. As the distance between two points in the product we take maximal distance between coordinates of these points. In the main theorem we show the homeomorphism between [:${\cal E}^i_{\rm T}, {\cal E}^j_{\rm T}$:] and ${\cal E}^{i+j}_{\rm T}$.

- Katsumi Wasaki, Noboru Endou.
Full Subtracter Circuit. Part I,
Formalized Mathematics 8(1), pages 77-81, 1999. MML Identifier: FSCIRC_1

**Summary**: We formalize the concept of the full subtracter circuit, define the structures of bit subtract/borrow units for binary operations, and prove the stability of the circuit.

- Yuguang Yang, Katsumi Wasaki, Yasushi Fuwa, Yatsuka Nakamura.
Correctness of Binary Counter Circuits,
Formalized Mathematics 8(1), pages 83-85, 1999. MML Identifier: GATE_2

**Summary**: This article introduces the verification of the correctness for the operations and the specification of the 3-bit counter. Both cases: without reset input and with reset input are considered. The proof was proposed by Y. Nakamura in \cite{GATE_1.ABS}.

- Yuguang Yang, Katsumi Wasaki, Yasushi Fuwa, Yatsuka Nakamura.
Correctness of Johnson Counter Circuits,
Formalized Mathematics 8(1), pages 87-91, 1999. MML Identifier: GATE_3

**Summary**: This article introduces the verification of the correctness for the operations and the specification of the Johnson counter. We formalize the concepts of 2-bit, 3-bit and 4-bit Johnson counter circuits with a reset input, and define the specification of the state transitions without the minor loop.

- Noboru Endou, Artur Kornilowicz.
The Definition of the Riemann Definite Integral and some Related Lemmas,
Formalized Mathematics 8(1), pages 93-102, 1999. MML Identifier: INTEGRA1

**Summary**: This article introduces the Riemann definite integral on the closed interval of real. We present the definitions and related lemmas of the closed interval. We formalize the concept of the Riemann definite integral and the division of the closed interval of real, and prove the additivity of the integral.

- Takashi Mitsuishi, Yuguang Yang.
Properties of the Trigonometric Function,
Formalized Mathematics 8(1), pages 103-106, 1999. MML Identifier: SIN_COS2

**Summary**: This article introduces the monotone increasing and the monotone decreasing of {\em sinus} and {\em cosine}, and definitions of hyperbolic {\em sinus}, hyperbolic {\em cosine} and hyperbolic {\em tangent}, and some related formulas about them.

- Shunichi Kobayashi, Yatsuka Nakamura.
Predicate Calculus for Boolean Valued Functions. Part II,
Formalized Mathematics 8(1), pages 107-109, 1999. MML Identifier: BVFUNC_4

**Summary**: In this paper, we have proved some elementary predicate calculus formulae containing the quantifiers of Boolean valued functions with respect to partitions. Such a theory is an analogy of usual predicate logic.

- Shunichi Kobayashi, Yatsuka Nakamura.
Propositional Calculus for Boolean Valued Functions. Part I,
Formalized Mathematics 8(1), pages 111-113, 1999. MML Identifier: BVFUNC_5

**Summary**: In this paper, we have proved some elementary propositional calculus formulae for Boolean valued functions.

- Shunichi Kobayashi, Yatsuka Nakamura.
Propositional Calculus for Boolean Valued Functions. Part II,
Formalized Mathematics 8(1), pages 115-117, 1999. MML Identifier: BVFUNC_6

**Summary**: In this paper, we have proved some elementary propositional calculus formulae for Boolean valued functions.

- Jing-Chao Chen.
Insert Sort on \SCMFSA,
Formalized Mathematics 8(1), pages 119-127, 1999. MML Identifier: SCMISORT

**Summary**: This article describes the insert sorting algorithm using macro instructions such as if-Macro (conditional branch macro instructions), for-loop macro instructions and While-Macro instructions etc. From the viewpoint of initialization, we generalize the halting and computing problem of the While-Macro. Generally speaking, it is difficult to judge whether the While-Macro is halting or not by way of loop inspection. For this reason, we introduce a practical and simple method, called body-inspection. That is, in many cases, we can prove the halting problem of the While-Macro by only verifying the nature of the body of the While-Macro, rather than the While-Macro itself. In fact, we have used this method in justifying the halting of the insert sorting algorithm. Finally, we prove that the insert sorting algorithm given in the article is autonomic and its computing result is correct.

- Yuguang Yang, Katsumi Wasaki, Yasushi Fuwa, Yatsuka Nakamura.
Correctness of a Cyclic Redundancy Check Code Generator,
Formalized Mathematics 8(1), pages 129-132, 1999. MML Identifier: GATE_4

**Summary**: We prove the correctness of the division circuit and the CRC (cyclic redundancy checks) circuit by verifying the contents of the register after one shift. Circuits with 12-bit register and 16-bit register are taken as examples. All the proofs are done formally.

- Andrzej Trybulec.
Defining by Structural Induction in the Positive Propositional Language,
Formalized Mathematics 8(1), pages 133-137, 1999. MML Identifier: HILBERT2

**Summary**: The main goal of the paper consists in proving schemes for defining by structural induction in the language defined by Adam Grabowski \cite{HILBERT1.ABS}. The article consists of four parts. Besides the preliminaries where we prove some simple facts still missing in the library, they are: \item{-} ``About the language'' in which the consequences of the fact that the algebra of formulae is free are formulated, \item{-} ``Defining by structural induction'' in which two schemes are proved, \item{-} ``The tree of the subformulae'' in which a scheme proved in the previous section is used to define the tree of subformulae; also some simple facts about the tree are proved.

- Czeslaw Bylinski.
Some Properties of Cells on Go-Board,
Formalized Mathematics 8(1), pages 139-146, 1999. MML Identifier: GOBRD13

**Summary**:

- Shunichi Kobayashi.
Propositional Calculus for Boolean Valued Functions. Part III,
Formalized Mathematics 8(1), pages 147-148, 1999. MML Identifier: BVFUNC_7

**Summary**: In this paper, we have proved some elementary propositional calculus formulae for Boolean valued functions.

- Shunichi Kobayashi.
Propositional Calculus for Boolean Valued Functions. Part IV,
Formalized Mathematics 8(1), pages 149-150, 1999. MML Identifier: BVFUNC_8

**Summary**: In this paper, we have proved some elementary propositional calculus formulae for Boolean valued functions.

- Akihiko Uchibori, Noboru Endou.
Basic Properties of Genetic Algorithm,
Formalized Mathematics 8(1), pages 151-160, 1999. MML Identifier: GENEALG1

**Summary**: We defined the set of the gene, the space treated by the genetic algorithm and the individual of the space. Moreover, we defined some genetic operators such as one point crossover and two points crossover, and the validity of many characters were proven.

- Shunichi Kobayashi.
Propositional Calculus for Boolean Valued Functions. Part V,
Formalized Mathematics 8(1), pages 161-162, 1999. MML Identifier: BVFUNC_9

**Summary**: In this paper, we have proved some elementary propositional calculus formulae for Boolean valued functions.

- Artur Kornilowicz.
Properties of Left and Right Components,
Formalized Mathematics 8(1), pages 163-168, 1999. MML Identifier: GOBRD14

**Summary**:

- Christoph Schwarzweller.
Noetherian Lattices,
Formalized Mathematics 8(1), pages 169-174, 1999. MML Identifier: LATTICE6

**Summary**: In this article we define noetherian and co-noetherian lattices and show how some properties concerning upper and lower neighbours, irreducibility and density can be improved when restricted to these kinds of lattices. In addition we define atomic lattices.

- Jing-Chao Chen.
A Small Computer Model with Push-Down Stack,
Formalized Mathematics 8(1), pages 175-182, 1999. MML Identifier: SCMPDS_1

**Summary**: The SCMFSA computer can prove the correctness of many algorithms. Unfortunately, it cannot prove the correctness of recursive algorithms. For this reason, this article improves the SCMFSA computer and presents a Small Computer Model with Push-Down Stack (called SCMPDS for short). In addition to conventional arithmetic and "goto" instructions, we increase two new instructions such as "return" and "save instruction-counter" in order to be able to design recursive programs.

- Jing-Chao Chen.
The SCMPDS Computer and the Basic Semantics of its Instructions,
Formalized Mathematics 8(1), pages 183-191, 1999. MML Identifier: SCMPDS_2

**Summary**: The article defines the SCMPDS computer and its instructions. The SCMPDS computer consists of such instructions as conventional arithmetic, ``goto'', ``return'' and ``save instruction-counter'' (``saveIC'' for short). The address used in the ``goto'' instruction is an offset value rather than a pointer in the standard sense. Thus, we don't define halting instruction directly but define it by ``goto 0'' instruction. The ``saveIC'' and ``return'' equal almost call and return statements in the usual high programming language. Theoretically, the SCMPDS computer can implement all algorithms described by the usual high programming language including recursive routine. In addition, we describe the execution semantics and halting properties of each instruction.

- Jing-Chao Chen.
Computation and Program Shift in the SCMPDS Computer,
Formalized Mathematics 8(1), pages 193-199, 1999. MML Identifier: SCMPDS_3

**Summary**: A finite partial state is said to be autonomic if the computation results in any two states containing it are same on its domain. On the basis of this definition, this article presents some computation results about autonomic finite partial states of the SCMPDS computer. Because the instructions of the SCMPDS computer are more complicated than those of the SCMFSA computer, the results given by this article are weaker than those reported previously by the article on the SCMFSA computer. The second task of this article is to define the notion of program shift. The importance of this notion is that the computation of some program blocks can be simplified by shifting a program block to the initial position.

- Jing-Chao Chen.
The Construction and Shiftability of Program Blocks for SCMPDS,
Formalized Mathematics 8(1), pages 201-210, 1999. MML Identifier: SCMPDS_4

**Summary**: In this article, a program block is defined as a finite sequence of instructions stored consecutively on initial positions. Based on this definition,any program block with more than two instructions can be viewed as the combination of two smaller program blocks. To describe the computation of a program block by the result of its two sub-blocks, we introduce the notions of paraclosed, parahalting, valid, and shiftable, the meaning of which may be stated as follows: \begin{itemize} \item[-] a program is paraclosed if and only if any state containing it is closed, \item[-] a program is parahalting if and only if any state containing it is halting, \item[-] in a program block, a jumping instruction is valid if its jumping offset is valid, \item[-] a program block is shiftable if it does not contain any return and saveIC instructions, and each instruction in it is valid. \end{itemize} When a program block is shiftable, its computing result does not depend on its storage position.

- Jing-Chao Chen.
Computation of Two Consecutive Program Blocks for SCMPDS,
Formalized Mathematics 8(1), pages 211-217, 1999. MML Identifier: SCMPDS_5

**Summary**: In this article, a program block without halting instructions is called No-StopCode program block. If a program consists of two blocks, where the first block is parahalting (i.e. halt for all states) and No-StopCode, and the second block is parahalting and shiftable, it can be computed by combining the computation results of the two blocks. For a program which consists of a instruction and a block, we obtain a similar conclusion. For a large amount of programs, the computation method given in the article is useful, but it is not suitable to recursive programs.

- Jing-Chao Chen.
The Construction and Computation of Conditional Statements for SCMPDS,
Formalized Mathematics 8(1), pages 219-234, 1999. MML Identifier: SCMPDS_6

**Summary**: We construct conditional statements like the usual high level program language by program blocks of SCMPDS. Roughly speaking, the article justifies such a fact that when the condition of a conditional statement is true (false), and the true (false) branch is shiftable, parahalting and does not contain any halting instruction, and the false branch is shiftable, then it is halting and its computation result equals that of the true (false) branch. The parahalting means some program halts for all states, this is strong condition. For this reason, we introduce the notions of "is\_closed\_on" and "is\_halting\_on". The predicate "A is\_closed\_on B" denotes program A is closed on state B, and "A is\_halting\_on B" denotes program A is halting on state B. We obtain a similar theorem to the above fact by replacing parahalting by "is\_closed\_on" and "is\_halting\_on".

- Yatsuka Nakamura, Andrzej Trybulec, Czeslaw Bylinski.
Bounded Domains and Unbounded Domains,
Formalized Mathematics 8(1), pages 1-13, 1999. MML Identifier: JORDAN2C
**Volume 9, 2001**- Number 1
- Jing-Chao Chen.
Recursive Euclide Algorithm,
Formalized Mathematics 9(1), pages 1-4, 2001. MML Identifier: SCMP_GCD

**Summary**: The earlier SCM computer did not contain recursive function, so Trybulec and Nakamura proved the correctness of the Euclid's algorithm only by way of an iterative program. However, the recursive method is a very important programming method, furthermore, for some algorithms, for example Quicksort, only by employing a recursive method (note push-down stack is essentially also a recursive method) can they be implemented. The main goal of the article is to test the recursive function of the SCMPDS computer by proving the correctness of the Euclid's algorithm by way of a recursive program. In this article, we observed that the memory required by the recursive Euclide algorithm is variable but it is still autonomic. Although the algorithm here is more complicated than the non-recursive algorithm, its focus is that the SCMPDS computer will be able to implement many algorithms like Quicksort which the SCM computer cannot do.

- Adam Grabowski.
Scott-Continuous Functions. Part II,
Formalized Mathematics 9(1), pages 5-11, 2001. MML Identifier: WAYBEL24

**Summary**:

- Jaroslaw Gryko, Artur Kornilowicz.
Some Properties of Isomorphism between Relational Structures. On the Product of Topological Spaces,
Formalized Mathematics 9(1), pages 13-18, 2001. MML Identifier: YELLOW14

**Summary**:

- Czeslaw Bylinski, Mariusz Zynel.
Cages -- the External Approximation of Jordan's Curve,
Formalized Mathematics 9(1), pages 19-24, 2001. MML Identifier: JORDAN9

**Summary**: On the Euclidean plane Jordan's curve may be approximated with a polygonal path of sides parallel to coordinate axes, either externally, or internally. The paper deals with the external approximation, and the existence of a {\em Cage} -- an external polygonal path -- is proved.

- Robert Milewski.
Components and Basis of Topological Spaces,
Formalized Mathematics 9(1), pages 25-29, 2001. MML Identifier: YELLOW15

**Summary**: This article contains many facts about components and basis of topological spaces.

- Artur Kornilowicz.
Properties of the External Approximation of Jordan's Curve,
Formalized Mathematics 9(1), pages 31-34, 2001. MML Identifier: JORDAN10

**Summary**:

- Freek Wiedijk.
Irrationality of $e$,
Formalized Mathematics 9(1), pages 35-39, 2001. MML Identifier: IRRAT_1

**Summary**: We prove the irrationality of square roots of prime numbers and of the number $e$. In order to be able to prove the last, a proof is given that {\tt number\_e = exp(1)} as defined in the Mizar library, that is that $$\lim_{n\rightarrow\infty} (1+{1\over n})^n = \sum_{k=0}^\infty {1\over k!}$$

- Artur Kornilowicz, Jaroslaw Gryko.
Injective Spaces. Part II,
Formalized Mathematics 9(1), pages 41-47, 2001. MML Identifier: WAYBEL25

**Summary**:

- Shunichi Kobayashi.
Propositional Calculus for Boolean Valued Functions. Part VI,
Formalized Mathematics 9(1), pages 49-50, 2001. MML Identifier: BVFUNC10

**Summary**: In this paper, we proved some elementary propositional calculus formulae for Boolean valued functions.

- Shunichi Kobayashi, Yatsuka Nakamura.
Predicate Calculus for Boolean Valued Functions. Part III,
Formalized Mathematics 9(1), pages 51-53, 2001. MML Identifier: BVFUNC11

**Summary**: In this paper, we proved some elementary predicate calculus formulae containing the quantifiers of Boolean valued functions with respect to partitions. Such a theory is an analogy of usual predicate logic.

- Christoph Schwarzweller.
A Characterization of Concept Lattices. Dual Concept Lattices,
Formalized Mathematics 9(1), pages 55-59, 2001. MML Identifier: CONLAT_2

**Summary**: In this article we continue the formalization of concept lattices following \cite{GanterWille}. We give necessary and sufficient conditions for a complete lattice to be isomorphic to a given formal context. As a by-product we get that a lattice is complete if and only if it is isomorphic to a concept lattice. In addition we introduce dual formal concepts and dual concept lattices and prove that the dual of a concept lattice over a formal context is isomorphic to the concept lattice over the dual formal context.

- Shunichi Kobayashi, Yatsuka Nakamura.
Predicate Calculus for Boolean Valued Functions. Part IV,
Formalized Mathematics 9(1), pages 61-63, 2001. MML Identifier: BVFUNC12

**Summary**:

- Shunichi Kobayashi, Yatsuka Nakamura.
Predicate Calculus for Boolean Valued Functions. Part V,
Formalized Mathematics 9(1), pages 65-70, 2001. MML Identifier: BVFUNC13

**Summary**: In this paper, we proved some elementary predicate calculus formulae containing the quantifiers of Boolean valued functions with respect to partitions. Such a theory is an analogy of usual predicate logic.

- Yoshinori Fujisawa, Yasushi Fuwa.
Definitions of Radix-$2^k$ Signed-Digit Number and its Adder Algorithm,
Formalized Mathematics 9(1), pages 71-75, 2001. MML Identifier: RADIX_1

**Summary**: In this article, a radix-$2^k$ signed-digit number (Radix-$2^k$ SD number) is defined and based on it a high-speed adder algorithm is discussed. \par The processes of coding and encoding for public-key cryptograms require a great deal of addition operations of natural number of many figures. This results in a~long time for the encoding and decoding processes. It is possible to reduce the processing time using the high-speed adder algorithm.\par In the first section of this article, we prepared some useful theorems for natural numbers and integers. In the second section, we defined the concept of radix-$2^k$, a set named $k$-SD and proved some properties about them. In the third section, we provide some important functions for generating Radix-$2^k$ SD numbers from natural numbers and natural numbers from Radix-$2^k$ SD numbers. In the fourth section, we defined the carry and data components of addition with Radix-$2^k$ SD numbers and some properties about them. In the fifth section, we defined a theorem for checking whether or not a natural number can be expressed as $n$ digits Radix-$2^k$ SD number. \par In the last section, a high-speed adder algorithm on Radix-$2^k$ SD numbers is proposed and we provided some properties. In this algorithm, the carry of each digit has an effect on only the next digit. Properties of the relationships of the results of this algorithm to the operations of natural numbers are also given.

- Grzegorz Bancerek.
Retracts and Inheritance,
Formalized Mathematics 9(1), pages 77-85, 2001. MML Identifier: YELLOW16

**Summary**:

- Grzegorz Bancerek.
Technical Preliminaries to Algebraic Specifications,
Formalized Mathematics 9(1), pages 87-93, 2001. MML Identifier: ALGSPEC1

**Summary**:

- Piotr Rudnicki, Andrzej Trybulec.
Multivariate Polynomials with Arbitrary Number of Variables,
Formalized Mathematics 9(1), pages 95-110, 2001. MML Identifier: POLYNOM1

**Summary**: The goal of this article is to define multivariate polynomials in arbitrary number of indeterminates and then to prove that they constitute a ring (over appropriate structure of coefficients).\par The introductory section includes quite a number of auxiliary lemmas related to many different parts of the MML. The second section characterizes the sequence flattening operation, introduced in \cite{DTCONSTR.ABS}, but so far lacking theorems about its fundamental properties.\par We first define formal power series in arbitrary number of variables. The auxiliary concept on which the construction of formal power series is based is the notion of a bag. A bag of a set $X$ is a natural function on $X$ which is zero almost everywhere. The elements of $X$ play the role of formal variables and a bag gives their exponents thus forming a power product. Series are defined for an ordered set of variables (we use ordinal numbers). A series in $o$ variables over a structure $S$ is a function assigning an element of the carrier of $S$ (coefficient) to each bag of $o$.\par We define the operations of addition, complement and multiplication for formal power series and prove their properties which depend on assumed properties of the structure from which the coefficients are taken. (We would like to note that proving associativity of multiplication turned out to be technically complicated.)\par Polynomial is defined as a formal power series with finite number of non zero coefficients. In conclusion, the ring of polynomials is defined.

- Grzegorz Bancerek.
Continuous Lattices of Maps between T$_0$ Spaces,
Formalized Mathematics 9(1), pages 111-117, 2001. MML Identifier: WAYBEL26

**Summary**: Formalization of \cite[pp. 128--130]{CCL}, chapter II, section 4 (4.1 -- 4.9).

- Shunichi Kobayashi.
Predicate Calculus for Boolean Valued Functions. Part VI,
Formalized Mathematics 9(1), pages 119-121, 2001. MML Identifier: BVFUNC14

**Summary**: In this paper, we proved some elementary predicate calculus formulae containing the quantifiers of Boolean valued functions with respect to partitions. Such a theory is an analogy of usual predicate logic.

- Shunichi Kobayashi.
Predicate Calculus for Boolean Valued Functions. Part VII,
Formalized Mathematics 9(1), pages 123-125, 2001. MML Identifier: BVFUNC15

**Summary**:

- Shunichi Kobayashi.
Predicate Calculus for Boolean Valued Functions. Part VIII,
Formalized Mathematics 9(1), pages 127-129, 2001. MML Identifier: BVFUNC16

**Summary**:

- Shunichi Kobayashi.
Predicate Calculus for Boolean Valued Functions. Part IX,
Formalized Mathematics 9(1), pages 131-133, 2001. MML Identifier: BVFUNC17

**Summary**:

- Richard Krueger, Piotr Rudnicki, Paul Shelley.
Asymptotic Notation. Part I: Theory,
Formalized Mathematics 9(1), pages 135-142, 2001. MML Identifier: ASYMPT_0

**Summary**: The widely used textbook by Brassard and Bratley \cite{BraBra96} includes a chapter devoted to asymptotic notation (Chapter 3, pp. 79--97). We have attempted to test how suitable the current version of Mizar is for recording this type of material in its entirety. A more detailed report on this experiment will be available separately. This article presents the development of notions and a follow-up article \cite{ASYMPT_1.ABS} includes examples and solutions to problems. The preliminaries introduce a number of properties of real sequences, some operations on real sequences, and a characterization of convergence. The remaining sections in this article correspond to sections of Chapter 3 of \cite{BraBra96}. Section 2 defines the $O$ notation and proves the threshold, maximum, and limit rules. Section 3 introduces the $\Omega$ and $\Theta$ notations and their analogous rules. Conditional asymptotic notation is defined in Section 4 where smooth functions are also discussed. Section 5 defines some operations on asymptotic notation (we have decided not to introduce the asymptotic notation for functions of several variables as it is a straightforward generalization of notions for unary functions).

- Richard Krueger, Piotr Rudnicki, Paul Shelley.
Asymptotic Notation. Part II: Examples and Problems,
Formalized Mathematics 9(1), pages 143-154, 2001. MML Identifier: ASYMPT_1

**Summary**: The widely used textbook by Brassard and Bratley \cite{BraBra96} includes a chapter devoted to asymptotic notation (Chapter 3, pp. 79--97). We have attempted to test how suitable the current version of Mizar is for recording this type of material in its entirety. This article is a follow-up to \cite{ASYMPT_0.ABS} in which we introduced the basic notions and general theory. This article presents a Mizar formalization of examples and solutions to problems from Chapter 3 of \cite{BraBra96} (some of the examples and solved problems are also in \cite{ASYMPT_0.ABS}). Not all problems have been solved as some required solutions not amenable for formalization.

- Shunichi Kobayashi.
Predicate Calculus for Boolean Valued Functions. Part X,
Formalized Mathematics 9(1), pages 155-156, 2001. MML Identifier: BVFUNC18

**Summary**:

- Shunichi Kobayashi.
Predicate Calculus for Boolean Valued Functions. Part XI,
Formalized Mathematics 9(1), pages 157-159, 2001. MML Identifier: BVFUNC19

**Summary**:

- Shunichi Kobayashi.
Four Variable Predicate Calculus for Boolean Valued Functions. Part I,
Formalized Mathematics 9(1), pages 161-165, 2001. MML Identifier: BVFUNC20

**Summary**:

- Shunichi Kobayashi.
Four Variable Predicate Calculus for Boolean Valued Functions. Part II,
Formalized Mathematics 9(1), pages 167-170, 2001. MML Identifier: BVFUNC21

**Summary**:

- Grzegorz Bancerek, Adam Naumowicz.
Function Spaces in the Category of Directed Suprema Preserving Maps,
Formalized Mathematics 9(1), pages 171-177, 2001. MML Identifier: WAYBEL27

**Summary**: Formalization of \cite[pp. 115--117]{ccl}, chapter II, section 2 (2.5 -- 2.10).

- Takashi Mitsuishi, Katsumi Wasaki, Yasunari Shidama.
Property of Complex Functions,
Formalized Mathematics 9(1), pages 179-184, 2001. MML Identifier: CFUNCT_1

**Summary**: This article introduces properties of complex function, calculations of them, boundedness and constant.

- Takashi Mitsuishi, Katsumi Wasaki, Yasunari Shidama.
Property of Complex Sequence and Continuity of Complex Function,
Formalized Mathematics 9(1), pages 185-190, 2001. MML Identifier: CFCONT_1

**Summary**: This article introduces properties of complex sequence and continuity of complex function. The first section shows convergence of complex sequence and constant complex sequence. In the next section, definition of continuity of complex function and properties of continuous complex function are shown.

- Noboru Endou, Katsumi Wasaki, Yasunari Shidama.
Scalar Multiple of Riemann Definite Integral,
Formalized Mathematics 9(1), pages 191-196, 2001. MML Identifier: INTEGRA2

**Summary**: The goal of this article is to prove a scalar multiplicity of Riemann definite integral. Therefore, we defined a scalar product to the subset of real space, and we proved some relating lemmas. At last, we proved a scalar multiplicity of Riemann definite integral. As a result, a linearity of Riemann definite integral was proven by unifying the previous article \cite{INTEGRA1.ABS}.

- Noboru Endou, Katsumi Wasaki, Yasunari Shidama.
Darboux's Theorem,
Formalized Mathematics 9(1), pages 197-200, 2001. MML Identifier: INTEGRA3

**Summary**: In this article, we have proved the Darboux's theorem. This theorem is important to prove the Riemann integrability. We can replace an upper bound and a lower bound of a function which is the definition of Riemann integration with convergence of sequence by Darboux's theorem.

- Shunichi Kobayashi.
Five Variable Predicate Calculus for Boolean Valued Functions. Part I,
Formalized Mathematics 9(1), pages 201-204, 2001. MML Identifier: BVFUNC22

**Summary**:

- Shunichi Kobayashi.
Six Variable Predicate Calculus for Boolean Valued Functions. Part I,
Formalized Mathematics 9(1), pages 205-208, 2001. MML Identifier: BVFUNC23

**Summary**:

- Jing-Chao Chen, Piotr Rudnicki.
The Construction and Computation of for-loop Programs for SCMPDS,
Formalized Mathematics 9(1), pages 209-219, 2001. MML Identifier: SCMPDS_7

**Summary**: This article defines two for-loop statements for SCMPDS. One is called for-up, which corresponds to ``for (i=x; i$<$0; i+=n) S'' in C language. Another is called for-down, which corresponds to ``for (i=x; i$>$0; i-=n) S''. Here, we do not present their unconditional halting (called parahalting) property, because we have not found that there exists a useful for-loop statement with unconditional halting, and the proof of unconditional halting is much simpler than that of conditional halting. It is hard to formalize all halting conditions, but some cases can be formalized. We choose loop invariants as halting conditions to prove halting problem of for-up/down statements. When some variables (except the loop control variable) keep undestroyed on a set for the loop invariant, and the loop body is halting for this condition, the corresponding for-up/down is halting and computable under this condition. The computation of for-loop statements can be realized by evaluating its body. At the end of the article, we verify for-down statements by two examples for summing.

- Shunichi Kobayashi.
Predicate Calculus for Boolean Valued Functions. Part XII,
Formalized Mathematics 9(1), pages 221-235, 2001. MML Identifier: BVFUNC24

**Summary**: In this paper, we proved some elementary predicate calculus formulae containing the quantifiers of Boolean valued functions with respect to partitions. Such a theory is an analogy of ordinary predicate logic.

- Jing-Chao Chen.
Recursive Euclide Algorithm,
Formalized Mathematics 9(1), pages 1-4, 2001. MML Identifier: SCMP_GCD
- Number 2
- Bartlomiej Skorulski.
Lim-Inf Convergence,
Formalized Mathematics 9(2), pages 237-240, 2001. MML Identifier: WAYBEL28

**Summary**: This work continues the formalization of \cite{CCL}. Theorems from Chapter III, Section 3, pp. 158--159 are proved.

- Grzegorz Bancerek, Adam Naumowicz.
The Characterization of the Continuity of Topologies,
Formalized Mathematics 9(2), pages 241-247, 2001. MML Identifier: WAYBEL29

**Summary**: Formalization of \cite[pp. 128--130]{CCL}, chapter II, section 4 (4.10, 4.11).

- Artur Kornilowicz.
Meet Continuous Lattices Revisited,
Formalized Mathematics 9(2), pages 249-254, 2001. MML Identifier: WAYBEL30

**Summary**: This work is a continuation of formalization of \cite{CCL}. Theorems from Chapter III, Section 2, pp. 153--156 are proved.

- Robert Milewski.
Weights of Continuous Lattices,
Formalized Mathematics 9(2), pages 255-259, 2001. MML Identifier: WAYBEL31

**Summary**: This work is a continuation of formalization of \cite{CCL}. Theorems from Chapter III, Section 4, pp. 170--171 are proved.

- Marek Dudzicz.
Representation Theorem for Finite Distributive Lattices,
Formalized Mathematics 9(2), pages 261-264, 2001. MML Identifier: LATTICE7

**Summary**: In the article the representation theorem for finite distributive lattice as rings of sets is presented. Auxiliary concepts are introduced. Namely, the concept of the height of an element, the maximal element in a chain, immediate predecessor of an element and ring of sets. Besides the scheme of induction in finite lattice is proved.

- Anna Justyna Milewska.
The Field of Complex Numbers,
Formalized Mathematics 9(2), pages 265-269, 2001. MML Identifier: COMPLFLD

**Summary**: This article contains the definition and many facts about the field of complex numbers.

- Noboru Endou, Katsumi Wasaki, Yasunari Shidama.
Integrability of Bounded Total Functions,
Formalized Mathematics 9(2), pages 271-274, 2001. MML Identifier: INTEGRA4

**Summary**: All these results have been obtained by Darboux's theorem in our previous article \cite{INTEGRA3.ABS}. In addition, we have proved the first mean value theorem to Riemann integral.

- Yasushi Fuwa, Yoshinori Fujisawa.
High-Speed Algorithms for RSA Cryptograms,
Formalized Mathematics 9(2), pages 275-279, 2001. MML Identifier: RADIX_2

**Summary**: In this article, we propose a new high-speed processing method for encoding and decoding the RSA cryptogram that is a kind of public-key cryptogram. This cryptogram is not only used for encrypting data, but also for such purposes as authentication. However, the encoding and decoding processes take a long time because they require a great deal of calculations. As a result, this cryptogram is not suited for practical use. Until now, we proposed a high-speed algorithm of addition using radix-$2^k$ signed-digit numbers and clarified correctness of it (\cite{RADIX_1.ABS}). In this article, we defined two new operations for a high-speed coding and encoding processes on public-key cryptograms based on radix-$2^k$ signed-digit (SD) numbers. One is calculation of $(a*b)$ mod $c$ ($a,b,c$ are natural numbers). Another one is calculation of $(a^b)$ mod $c$ ($a,b,c$ are natural numbers). Their calculations are realized repetition of addition. We propose a high-speed algorithm of their calculations using proposed addition algorithm and clarify the correctness of them. In the first section, we prepared some useful theorems for natural numbers and integers and so on. In the second section, we proved some properties of addition operation using a radix-$2^k$ SD numbers. In the third section, we defined some functions on the relation between a finite sequence of k-SD and a finite sequence of ${\Bbb N}$ and proved some properties about them. In the fourth section, algorithm of calculation of $(a*b)$ mod $c$ based on radix-$2^k$ SD numbers is proposed and its correctness is clarified. In the last section, algorithm of calculation of $(a^b)$ mod $c$ based on radix-$2^k$ SD numbers is proposed and we clarified its correctness.

- Noboru Endou, Katsumi Wasaki, Yasunari Shidama.
Definition of Integrability for Partial Functions from $\Bbb R$ to $\Bbb R$ and Integrability for Continuous Functions,
Formalized Mathematics 9(2), pages 281-284, 2001. MML Identifier: INTEGRA5

**Summary**: In this article, we defined the Riemann definite integral of partial function from ${\Bbb R}$ to ${\Bbb R}$. Then we have proved the integrability for the continuous function and differentiable function. Moreover, we have proved an elementary theorem of calculus.

- Noboru Endou, Katsumi Wasaki, Yasunari Shidama.
Introduction to Several Concepts of Convexity and Semicontinuity for Function from $\Bbb R$ to $\Bbb R$,
Formalized Mathematics 9(2), pages 285-289, 2001. MML Identifier: RFUNCT_4

**Summary**: This article is an introduction to convex analysis. In the beginning, we have defined the concept of strictly convexity and proved some basic properties between convexity and strictly convexity. Moreover, we have defined concepts of other convexity and semicontinuity, and proved their basic properties.

- Andrzej Trybulec, Piotr Rudnicki, Artur Kornilowicz.
Standard Ordering of Instruction Locations,
Formalized Mathematics 9(2), pages 291-301, 2001. MML Identifier: AMISTD_1

**Summary**:

- Artur Kornilowicz.
On the Composition of Macro Instructions of Standard Computers,
Formalized Mathematics 9(2), pages 303-316, 2001. MML Identifier: AMISTD_2

**Summary**:

- Artur Kornilowicz.
The Properties of Instructions of SCM over Ring,
Formalized Mathematics 9(2), pages 317-322, 2001. MML Identifier: SCMRING3

**Summary**:

- Josef Urban.
Basic Facts about Inaccessible and Measurable Cardinals,
Formalized Mathematics 9(2), pages 323-329, 2001. MML Identifier: CARD_FIL

**Summary**: Inaccessible, strongly inaccessible and measurable cardinals are defined, and it is proved that a measurable cardinal is strongly inaccessible. Filters on sets are defined, some facts related to the section about cardinals are proved. Existence of the Ulam matrix on non-limit cardinals is proved.

- Christoph Schwarzweller, Andrzej Trybulec.
The Evaluation of Multivariate Polynomials,
Formalized Mathematics 9(2), pages 331-338, 2001. MML Identifier: POLYNOM2

**Summary**:

- Robert Milewski.
The Ring of Polynomials,
Formalized Mathematics 9(2), pages 339-346, 2001. MML Identifier: POLYNOM3

**Summary**:

- Xiquan Liang.
Solving Roots of Polynomial Equations of Degree 2 and 3 with Real Coefficients,
Formalized Mathematics 9(2), pages 347-350, 2001. MML Identifier: POLYEQ_1

**Summary**: In this paper, we describe the definition of the first, second, and third degree algebraic equations and their properties. In Section 1, we defined the simple first-degree and second-degree (quadratic) equation and discussed the relation between the roots of each equation and their coefficients. Also, we clarified the form of the root within the range of real numbers. Furthermore, the extraction of the root using the discriminant of equation is clarified. In Section 2, we defined the third-degree (cubic) equation and clarified the relation between the three roots of this equation and its coefficient. Also, the form of these roots for various conditions is discussed. This solution is known as the Cardano solution.

- Takashi Mitsuishi, Noboru Endou, Yasunari Shidama.
The Concept of Fuzzy Set and Membership Function and Basic Properties of Fuzzy Set Operation,
Formalized Mathematics 9(2), pages 351-356, 2001. MML Identifier: FUZZY_1

**Summary**: This article introduces the fuzzy theory. At first, the definition of fuzzy set characterized by membership function is described. Next, definitions of empty fuzzy set and universal fuzzy set and basic operations of these fuzzy sets are shown. At last, exclusive sum and absolute difference which are special operation are introduced.

- Takashi Mitsuishi, Katsumi Wasaki, Yasunari Shidama.
Basic Properties of Fuzzy Set Operation and Membership Function,
Formalized Mathematics 9(2), pages 357-362, 2001. MML Identifier: FUZZY_2

**Summary**: This article introduces the fuzzy theory. The definition of the difference set, algebraic product and algebraic sum of fuzzy set is shown. In addition, basic properties of those operations are described. Basic properties of fuzzy set are a~little different from those of crisp set.

- Anna Justyna Milewska.
The Hahn Banach Theorem in the Vector Space over the Field of Complex Numbers,
Formalized Mathematics 9(2), pages 363-371, 2001. MML Identifier: HAHNBAN1

**Summary**: This article contains the Hahn Banach theorem in the vector space over the field of complex numbers.

- Bartlomiej Skorulski.
The Tichonov Theorem,
Formalized Mathematics 9(2), pages 373-376, 2001. MML Identifier: YELLOW17

**Summary**:

- Ewa Grdzka.
On the Order-consistent Topology of Complete and Uncomplete Lattices,
Formalized Mathematics 9(2), pages 377-382, 2001. MML Identifier: WAYBEL32

**Summary**: This paper is a continuation of the formalisation of \cite{CCL} pp.~108--109. Order-consistent and upper topologies are defined. The theorem that the Scott and the upper topologies are order-consistent is proved. Remark 1.4 and example 1.5(2) are generalized for proving this theorem.

- Adam Naumowicz.
On Segre's Product of Partial Line Spaces,
Formalized Mathematics 9(2), pages 383-390, 2001. MML Identifier: PENCIL_1

**Summary**: In this paper the concept of partial line spaces is presented. We also construct the Segre's product for a family of partial line spaces indexed by an arbitrary nonempty set.

- Robert Milewski.
The Evaluation of Polynomials,
Formalized Mathematics 9(2), pages 391-395, 2001. MML Identifier: POLYNOM4

**Summary**:

- Jing-Chao Chen.
The Construction and Computation of While-Loop Programs for SCMPDS,
Formalized Mathematics 9(2), pages 397-405, 2001. MML Identifier: SCMPDS_8

**Summary**: This article defines two while-loop statements on SCMPDS, i.e. ``while$<$0'' and ``while$>$0'', which resemble the while-statements of the common high language such as C. We previously presented a number of tricks for computing while-loop statements on SCMFSA, e.g. step-while. However, after inspecting a few realistic examples, we found that they are neither very useful nor of generalization. To cover much more computation cases of while-loop statements, we generalize the computation model of while-loop statements, based on the principle of Hoare's axioms on the verification of programs.

- Jing-Chao Chen.
Insert Sort on SCMPDS,
Formalized Mathematics 9(2), pages 407-412, 2001. MML Identifier: SCPISORT

**Summary**: The goal of this article is to examine the effectiveness of ``for-loop'' and ``while-loop'' statements on SCMPDS by insert sort. In this article, first of all, we present an approach to compute the execution result of ``for-loop'' program by ``loop-invariant'', based on Hoare's axioms for program verification. Secondly, we extend the fundamental properties of the finite sequence and complex instructions of SCMPDS. Finally, we prove the correctness of the insert sort program described in the article.

- Jing-Chao Chen.
Quick Sort on SCMPDS,
Formalized Mathematics 9(2), pages 413-418, 2001. MML Identifier: SCPQSORT

**Summary**: Proving the correctness of quick sort is much more complicated than proving the correctness of the insert sort. Its difficulty is determined mainly by the following points: \begin{itemize} \item Quick sort needs to use a push-down stack. \item It contains three nested loops. \item A subroutine of this algorithm, ``Partition'', has no loop-invariant. \end{itemize} This means we cannot justify the correctness of the ``Partition'' subroutine by the Hoare's axiom on program verification. This article is organized as follows. First, we present several fundamental properties of ``while'' program and finite sequence. Second, we define the ``Partition'' subroutine on SCMPDS, the task of which is to split a sequence into a smaller and a larger subsequence. The definition of quick sort on SCMPDS follows. Finally, we describe the basic property of the ``Partition'' and quick sort, and prove their correctness.

- Jing-Chao Chen.
Justifying the Correctness of the Fibonacci Sequence and the Euclide Algorithm by Loop-Invariant,
Formalized Mathematics 9(2), pages 419-427, 2001. MML Identifier: SCPINVAR

**Summary**: If a loop-invariant exists in a loop program, computing its result by loop-invariant is simpler and easier than computing its result by the inductive method. For this purpose, the article describes the premise and the final computation result of the program such as ``while$<$0'', ``while$>$0'', ``while$<>$0'' by loop-invariant. To test the effectiveness of the computation method given in this article, by using loop-invariant of the loop programs mentioned above, we justify the correctness of the following three examples: Summing $n$ integers (used for testing ``while$>$0''), Fibonacci sequence (used for testing ``while$<$0''), Greatest Common Divisor, i.e. Euclide algorithm (used for testing ``while$<>$0'').

- Marta Pruszynska, Marek Dudzicz.
On the Isomorphism between Finite Chains,
Formalized Mathematics 9(2), pages 429-430, 2001. MML Identifier: ORDERS_4

**Summary**:

- Mariusz Lapinski.
The J\'onsson Theorem about the Representation of Modular Lattices,
Formalized Mathematics 9(2), pages 431-438, 2001. MML Identifier: LATTICE8

**Summary**: Formalization of \cite[pp. 192--199]{Gratzer}, chapter IV. Partition Lattices, theorem 8.

- Bartlomiej Skorulski.
Lim-Inf Convergence,
Formalized Mathematics 9(2), pages 237-240, 2001. MML Identifier: WAYBEL28
- Number 3
- Andrzej Trybulec.
The Canonical Formulae,
Formalized Mathematics 9(3), pages 441-447, 2001. MML Identifier: HILBERT3

**Summary**:

- Adam Grabowski.
The Incompleteness of the Lattice of Substitutions,
Formalized Mathematics 9(3), pages 449-454, 2001. MML Identifier: HEYTING3

**Summary**: In \cite{HEYTING3.ABS} we proved that the lattice of substitutions, as defined in \cite{SUBSTLAT.ABS}, is a Heyting lattice (i.e. it is pseudo-complemented and it has the zero element). We show that the lattice needs not to be complete. Obviously, the example has to be infinite, namely we can take the set of natural numbers as variables and a singleton as a set of constants. The incompleteness has been shown for lattices of substitutions defined in terms of \cite{LATTICES.ABS} and relational structures \cite{ORDERS_1.ABS}.

- Robert Milewski.
Trigonometric Form of Complex Numbers,
Formalized Mathematics 9(3), pages 455-460, 2001. MML Identifier: COMPTRIG

**Summary**:

- Robert Milewski.
Fundamental Theorem of Algebra,
Formalized Mathematics 9(3), pages 461-470, 2001. MML Identifier: POLYNOM5

**Summary**:

- Hiroshi Yamazaki, Yoshinori Fujisawa, Yatsuka Nakamura.
On Replace Function and Swap Function for Finite Sequences,
Formalized Mathematics 9(3), pages 471-474, 2001. MML Identifier: FINSEQ_7

**Summary**: In this article, we show the property of the Replace Function and the Swap Function of finite sequences. In the first section, we prepared some useful theorems for finite sequences. In the second section, we defined the Replace function and proved some theorems about the function. This function replaces an element of a sequence by another value. In the third section, we defined the Swap function and proved some theorems about the function. This function swaps two elements of a sequence. In the last section, we show the property of composed functions of the Replace Function and the Swap Function.

- Hiroshi Yamazaki, Katsumi Wasaki.
The Correctness of the High Speed Array Multiplier Circuits,
Formalized Mathematics 9(3), pages 475-479, 2001. MML Identifier: GATE_5

**Summary**: This article introduces the verification of the correctness for the operations and the specification of the high speed array multiplier. We formalize the concepts of 2-by-2 and 3-by-3 bit Plain array multiplier, 3-by-3 Wallace tree multiplier circuit, and show that outputs of the array multiplier are equivalent to outputs of normal (sequencial) multiplier.

- Andrzej Trybulec.
Some Lemmas for the Jordan Curve Theorem,
Formalized Mathematics 9(3), pages 481-484, 2001. MML Identifier: JCT_MISC

**Summary**: I present some miscellaneous simple facts that are still missing in the library. The only common feature is that, most of them, were needed as lemmas in the proof of the Jordan curve theorem.

- Josef Urban.
Mahlo and Inaccessible Cardinals,
Formalized Mathematics 9(3), pages 485-489, 2001. MML Identifier: CARD_LAR

**Summary**: This article contains basic ordinal topology: closed unbounded and stationary sets and necessary theorems about them, completness of the centered system of Clubs of $M$, Mahlo and strongly Mahlo cardinals, the proof that (strongly) Mahlo is (strongly) inaccessible, and the proof that Rank of strongly inaccessible is a model of ZF.

- Noboru Endou, Katsumi Wasaki, Yasunari Shidama.
Basic Properties of Extended Real Numbers,
Formalized Mathematics 9(3), pages 491-494, 2001. MML Identifier: EXTREAL1

**Summary**: We introduce product, quotient and absolute value, and we prove some basic properties of extended real numbers.

- Noboru Endou, Katsumi Wasaki, Yasunari Shidama.
Definitions and Basic Properties of Measurable Functions,
Formalized Mathematics 9(3), pages 495-500, 2001. MML Identifier: MESFUNC1

**Summary**: In this article we introduce some definitions concerning measurable functions and prove related properties.

- Artur Kornilowicz, Robert Milewski, Adam Naumowicz, Andrzej Trybulec.
Gauges and Cages. Part I,
Formalized Mathematics 9(3), pages 501-509, 2001. MML Identifier: JORDAN1A

**Summary**:

- Noboru Endou, Katsumi Wasaki, Yasunari Shidama.
Some Properties of Extended Real Numbers Operations: abs, min and max,
Formalized Mathematics 9(3), pages 511-516, 2001. MML Identifier: EXTREAL2

**Summary**: In this article, we extend some properties concerning real numbers to extended real numbers. Almost all properties included in this article are extended properties of other articles: \cite{AXIOMS.ABS}, \cite{REAL_1.ABS}, \cite{ABSVALUE.ABS}, \cite{SQUARE_1.ABS} and \cite{REAL_2.ABS}.

- Takashi Mitsuishi, Katsumi Wasaki, Yasunari Shidama.
The Concept of Fuzzy Relation and Basic Properties of its Operation,
Formalized Mathematics 9(3), pages 517-524, 2001. MML Identifier: FUZZY_3

**Summary**: This article introduces the fuzzy relation. This is the expansion of usual relation, and the value is given at the fuzzy value. At first, the definition of the fuzzy relation characterized by membership function is described. Next, the definitions of the zero relation and universe relation and basic operations of these relations are shown.

- Noboru Endou, Katsumi Wasaki, Yasunari Shidama.
The Measurability of Extended Real Valued Functions,
Formalized Mathematics 9(3), pages 525-529, 2001. MML Identifier: MESFUNC2

**Summary**: In this article we prove the measurablility of some extended real valued functions which are $f$+$g$, $f$\,--\,$g$ and so on. Moreover, we will define the simple function which are defined on the sigma field. It will play an important role for the Lebesgue integral theory.

- Robert Milewski, Andrzej Trybulec, Artur Kornilowicz, Adam Naumowicz.
Some Properties of Cells and Arcs,
Formalized Mathematics 9(3), pages 531-535, 2001. MML Identifier: JORDAN1B

**Summary**:

- Gang Liu, Yasushi Fuwa, Masayoshi Eguchi.
Formal Topological Spaces,
Formalized Mathematics 9(3), pages 537-543, 2001. MML Identifier: FINTOPO2

**Summary**: This article is divided into two parts. In the first part, we prove some useful theorems on finite topological spaces. In the second part, in order to consider a family of neighborhoods to a point in a space, we extend the notion of finite topological space and define a new topological space, which we call formal topological space. We show the relation between formal topological space struct ({\tt FMT\_Space\_Str}) and the {\tt TopStruct} by giving a function ({\tt NeighSp}). And the following notions are introduced in formal topological spaces: boundary, closure, interior, isolated point, connected point, open set and close set, then some basic facts concerning them are proved. We will discuss the relation between the formal topological space and the finite topological space in future work.

- Adam Grabowski, Artur Kornilowicz, Andrzej Trybulec.
Some Properties of Cells and Gauges,
Formalized Mathematics 9(3), pages 545-548, 2001. MML Identifier: JORDAN1C

**Summary**:

- Andrzej Trybulec, Yatsuka Nakamura.
Again on the Order on a Special Polygon,
Formalized Mathematics 9(3), pages 549-553, 2001. MML Identifier: SPRECT_5

**Summary**:

- Artur Kornilowicz, Robert Milewski.
Gauges and Cages. Part II,
Formalized Mathematics 9(3), pages 555-558, 2001. MML Identifier: JORDAN1D

**Summary**:

- Christoph Schwarzweller.
The Binomial Theorem for Algebraic Structures,
Formalized Mathematics 9(3), pages 559-564, 2001. MML Identifier: BINOM

**Summary**: In this paper we prove the well-known binomial theorem for algebraic structures. In doing so we tried to be as modest as possible concerning the algebraic properties of the underlying structure. Consequently, we proved the binomial theorem for ``commutative rings'' in which the existence of an inverse with respect to addition is replaced by a weaker property of cancellation.

- Jonathan Backer, Piotr Rudnicki, Christoph Schwarzweller.
Ring Ideals,
Formalized Mathematics 9(3), pages 565-582, 2001. MML Identifier: IDEAL_1

**Summary**: We introduce the basic notions of ideal theory in rings. This includes left and right ideals, (finitely) generated ideals and some operations on ideals such as the addition of ideals and the radical of an ideal. In addition we introduce linear combinations to formalize the well-known characterization of generated ideals. Principal ideal domains and Noetherian rings are defined. The latter development follows \cite{Becker93}, pages 144--145.

- Jonathan Backer, Piotr Rudnicki.
Hilbert Basis Theorem,
Formalized Mathematics 9(3), pages 583-589, 2001. MML Identifier: HILBASIS

**Summary**: We prove the Hilbert basis theorem following \cite{Becker93}, page 145. First we prove the theorem for the univariate case and then for the multivariate case. Our proof for the latter is slightly different than in \cite{Becker93}. As a base case we take the ring of polynomilas with no variables. We also prove that a polynomial ring with infinite number of variables is not Noetherian.

- Franz Merkl.
Dynkin's Lemma in Measure Theory,
Formalized Mathematics 9(3), pages 591-595, 2001. MML Identifier: DYNKIN

**Summary**: This article formalizes the proof of Dynkin's lemma in measure theory. Dynkin's lemma is a useful tool in measure theory and probability theory: it helps frequently to generalize a statement about all elements of a intersection-stable set system to all elements of the sigma-field generated by that system.

- Mariusz Giero, Roman Matuszewski.
Lower Tolerance. Preliminaries to Wroclaw Taxonomy,
Formalized Mathematics 9(3), pages 597-603, 2001. MML Identifier: TAXONOM1

**Summary**: The paper introduces some preliminary notions concerning the Wroclaw taxonomy according to \cite{MatTry77}. The classifications and tolerances are defined and considered w.r.t. sets and metric spaces. We prove theorems showing various classifications based on tolerances.

- Grzegorz Bancerek.
Concrete Categories,
Formalized Mathematics 9(3), pages 605-621, 2001. MML Identifier: YELLOW18

**Summary**: In the paper, we develop the notation of duality and equivalence of categories and concrete categories based on \cite{ALTCAT_1.ABS}. The development was motivated by the duality theory for continuous lattices (see \cite[p. 189]{CCL}), where we need to cope with concrete categories of lattices and maps preserving their properties. For example, the category {\it UPS} of complete lattices and directed suprema preserving maps; or the category {\it INF} of complete lattices and infima preserving maps. As the main result of this paper it is shown that every category is isomorphic to its concretization (the concrete category with the same objects). Some useful schemes to construct categories and functors are also presented.

- Andrzej Trybulec.
Classes of Independent Partitions,
Formalized Mathematics 9(3), pages 623-625, 2001. MML Identifier: PARTIT_2

**Summary**: The paper includes proofs of few theorems proved earlier by Shunichi Kobayashi in many different settings.

- Jozef Bialas, Yatsuka Nakamura.
Some Properties of Dyadic Numbers and Intervals,
Formalized Mathematics 9(3), pages 627-630, 2001. MML Identifier: URYSOHN2

**Summary**: The article is the second part of a paper proving the fundamental Urysohn Theorem concerning the existence of a real valued continuous function on a normal topological space. The paper is divided into two parts. In the first part, we introduce some definitions and theorems concerning properties of intervals; in the second we prove some of properties of dyadic numbers used in proving Urysohn Lemma.

- Jozef Bialas, Yatsuka Nakamura.
The Urysohn Lemma,
Formalized Mathematics 9(3), pages 631-636, 2001. MML Identifier: URYSOHN3

**Summary**: This article is the third part of a paper proving the fundamental Urysohn Theorem concerning the existence of a real valued continuous function on a normal topological space. The paper is divided into two parts. In the first part, we describe the construction of the function solving thesis of the Urysohn Lemma. The second part contains the proof of the Urysohn Lemma in normal space and the proof of the same theorem for compact space.

- Ewa Grdzka.
The Algebra of Polynomials,
Formalized Mathematics 9(3), pages 637-643, 2001. MML Identifier: POLYALG1

**Summary**: In this paper we define the algebra of formal power series and the algebra of polynomials over an arbitrary field and prove some properties of these structures. We also formulate and prove theorems showing some general properties of sequences. These preliminaries will be used for defining and considering linear functionals on the algebra of polynomials.

- Grzegorz Bancerek.
Circuit Generated by Terms and Circuit Calculating Terms,
Formalized Mathematics 9(3), pages 645-657, 2001. MML Identifier: CIRCTRM1

**Summary**: In the paper we investigate the dependence between the structure of circuits and sets of terms. Circuits in our terminology (see \cite{CIRCUIT1.ABS}) are treated as locally-finite many sorted algebras over special signatures. Such approach enables to formalize every real circuit. The goal of this investigation is to specify circuits by terms and, enentualy, to have methods of formal verification of real circuits. The following notation is introduced in this paper: \begin{itemize} \item structural equivalence of circuits, i.e. equivalence of many sorted signatures, \item embedding of a circuit into another one, \item similarity of circuits (a concept narrower than isomorphism of many sorted algebras over equivalent signatures), \item calculation of terms by a circuit according to an algebra, \item specification of circuits by terms and an algebra. \end{itemize}

- Andrzej Trybulec.
The Canonical Formulae,
Formalized Mathematics 9(3), pages 441-447, 2001. MML Identifier: HILBERT3
- Number 4
- Artur Kornilowicz.
On the Instructions of \SCM,
Formalized Mathematics 9(4), pages 659-663, 2001. MML Identifier: AMI_6

**Summary**:

- Artur Kornilowicz.
Input and Output of Instructions,
Formalized Mathematics 9(4), pages 665-671, 2001. MML Identifier: AMI_7

**Summary**:

- Artur Kornilowicz.
On the Instructions of \SCMFSA,
Formalized Mathematics 9(4), pages 673-679, 2001. MML Identifier: SCMFSA10

**Summary**:

- Adam Grabowski.
Robbins Algebras vs. Boolean Algebras,
Formalized Mathematics 9(4), pages 681-690, 2001. MML Identifier: ROBBINS1

**Summary**: In the early 1930s, Huntington proposed several axiom systems for Boolean algebras. Robbins slightly changed one of them and asked if the resulted system is still a basis for variety of Boolean algebras. The solution (afirmative answer) was given in 1996 by McCune with the help of automated theorem prover EQP/{\sc Otter}. Some simplified and restucturized versions of this proof are known. In our version of proof that all Robbins algebras are Boolean we use the results of McCune \cite{McCuneRob}, Huntington \cite{Huntington1}, \cite{Huntington2}, \cite{Huntington3} and Dahn \cite{DahnRob}.

- Noboru Endou, Takashi Mitsuishi, Keiji Ohkubo.
Properties of Fuzzy Relation,
Formalized Mathematics 9(4), pages 691-695, 2001. MML Identifier: FUZZY_4

**Summary**: In this article, we introduce four fuzzy relations and the composition, and some useful properties are shown by them. In section 2, the definition of converse relation $R^{-1}$ of fuzzy relation $R$ and properties concerning it are described. In the next section, we define the composition of the fuzzy relation and show some properties. In the final section we describe the identity relation, the universe relation and the zero relation.

- Yatsuka Nakamura.
On Outside Fashoda Meet Theorem,
Formalized Mathematics 9(4), pages 697-704, 2001. MML Identifier: JGRAPH_2

**Summary**: We have proven the ``Fashoda Meet Theorem'' in \cite{JGRAPH_1.ABS}. Here we prove the outside version of it. It says that if Britain and France intended to set the courses for ships to the opposite side of Africa, they must also meet.

- Grzegorz Bancerek, Piotr Rudnicki.
The Set of Primitive Recursive Functions,
Formalized Mathematics 9(4), pages 705-720, 2001. MML Identifier: COMPUT_1

**Summary**: We follow \cite{Uspenski60} in defining the set of primitive recursive functions. The important helper notion is the homogeneous function from finite sequences of natural numbers into natural numbers where homogeneous means that all the sequences in the domain are of the same length. The set of all such functions is then used to define the notion of a set closed under composition of functions and under primitive recursion. We call a set primitively recursively closed iff it contains the initial functions (nullary constant function returning 0, unary successor and projection functions for all arities) and is closed under composition and primitive recursion. The set of primitive recursive functions is then defined as the smallest set of functions which is primitive recursively closed. We show that this set can be obtained by primitive recursive approximation. We finish with showing that some simple and well known functions are primitive recursive.

- Jing-Chao Chen, Yatsuka Nakamura.
Introduction to Turing Machines,
Formalized Mathematics 9(4), pages 721-732, 2001. MML Identifier: TURING_1

**Summary**: A Turing machine can be viewed as a simple kind of computer, whose operations are constrainted to reading and writing symbols on a tape, or moving along the tape to the left or right. In theory, one has proven that the computability of Turing machines is equivalent to recursive functions. This article defines and verifies the Turing machines of summation and three primitive functions which are successor, zero and project functions. It is difficult to compute sophisticated functions by simple Turing machines. Therefore, we define the combination of two Turing machines.

- Grzegorz Bancerek, Noboru Endou, Yuji Sakai.
On the Characterizations of Compactness,
Formalized Mathematics 9(4), pages 733-738, 2001. MML Identifier: YELLOW19

**Summary**: In the paper we show equivalence of the convergence of filters on a topological space and the convergence of nets in the space. We also give, five characterizations of compactness. Namely, for any topological space $T$ we proved that following condition are equivalent: \begin{itemize} \itemsep-3pt \item $T$ is compact, \item every ultrafilter on $T$ is convergent, \item every proper filter on $T$ has cluster point, \item every net in $T$ has cluster point, \item every net in $T$ has convergent subnet, \item every Cauchy net in $T$ is convergent. \end{itemize}

- Grzegorz Bancerek, Noboru Endou.
Compactness of Lim-inf Topology,
Formalized Mathematics 9(4), pages 739-743, 2001. MML Identifier: WAYBEL33

**Summary**: Formalization of \cite{CCL}, chapter III, section 3 (3.4--3.6).

- Grzegorz Bancerek.
Miscellaneous Facts about Functors,
Formalized Mathematics 9(4), pages 745-754, 2001. MML Identifier: YELLOW20

**Summary**: In the paper we show useful facts concerning reverse and inclusion functors and the restriction of functors. We also introduce a new notation for the intersection of categories and the isomorphism under arbitrary functors.

- Grzegorz Bancerek.
Categorial Background for Duality Theory,
Formalized Mathematics 9(4), pages 755-765, 2001. MML Identifier: YELLOW21

**Summary**: In the paper, we develop the notation of lattice-wise categories as concrete categories (see \cite{YELLOW18.ABS}) of lattices. Namely, the categories based on \cite{ALTCAT_1.ABS} with lattices as objects and at least monotone maps between them as morphisms. As examples, we introduce the categories {\it UPS}, {\it CONT}, and {\it ALG} with complete, continuous, and algebraic lattices, respectively, as objects and directed suprema preserving maps as morphisms. Some useful schemes to construct categories of lattices and functors between them are also presented.

- Grzegorz Bancerek.
Duality Based on the Galois Connection. Part I,
Formalized Mathematics 9(4), pages 767-778, 2001. MML Identifier: WAYBEL34

**Summary**: In the paper, we investigate the duality of categories of complete lattices and maps preserving suprema or infima according to \cite[p. 179--183; 1.1--1.12]{CCL}. The duality is based on the concept of the Galois connection.

- Grzegorz Bancerek, Artur Kornilowicz.
Yet Another Construction of Free Algebra,
Formalized Mathematics 9(4), pages 779-785, 2001. MML Identifier: MSAFREE3

**Summary**:

- Robert Milewski.
Upper and Lower Sequence of a Cage,
Formalized Mathematics 9(4), pages 787-790, 2001. MML Identifier: JORDAN1E

**Summary**:

- Barbara Dzienis.
On Polynomials with Coefficients in a Ring of Polynomials,
Formalized Mathematics 9(4), pages 791-794, 2001. MML Identifier: POLYNOM6

**Summary**: The main result of the paper is, that the ring of polynomials with $o_1$ variables and coefficients in the ring of polynomials with $o_2$ variables and coefficient in a ring $L$ is isomorphic with the ring with $o_1+o_2$ variables, and coefficients in $L$.

- Adam Naumowicz.
On Cosets in Segre's Product of Partial Linear Spaces,
Formalized Mathematics 9(4), pages 795-800, 2001. MML Identifier: PENCIL_2

**Summary**: This paper is a continuation of \cite{PENCIL_1.ABS}. We prove that the family of cosets in the Segre's product of partial linear spaces remains invariant under automorphisms.

- Yatsuka Nakamura.
On the Simple Closed Curve Property of the Circle and the Fashoda Meet Theorem,
Formalized Mathematics 9(4), pages 801-808, 2001. MML Identifier: JGRAPH_3

**Summary**: First, we prove the fact that the circle is the simple closed curve, which was defined as a curve homeomorphic to the square. For this proof, we introduce a mapping which is a homeomorphism from 2-dimensional plane to itself. This mapping maps the square to the circle. Secondly, we prove the Fashoda meet theorem for the circle using this homeomorphism.

- Freek Wiedijk.
Pythagorean Triples,
Formalized Mathematics 9(4), pages 809-812, 2001. MML Identifier: PYTHTRIP

**Summary**: A Pythagorean triple is a set of positive integers $\{ a,b,c \}$ with $a^2 + b^2 = c^2$. We prove that every Pythagorean triple is of the form $$a = n^2 - m^2 \qquad b = 2mn \qquad c = n^2 + m^2$$ or is a multiple of such a triple. Using this characterization we show that for every $n > 2$ there exists a Pythagorean triple $X$ with $n\in X$. Also we show that even the set of {\em simplified\/} Pythagorean triples is infinite.

- Adam Naumowicz.
Some Remarks on Finite Sequences on Go-boards,
Formalized Mathematics 9(4), pages 813-816, 2001. MML Identifier: JORDAN1F

**Summary**: This paper shows some properties of finite sequences on Go-boards. It also provides the partial correspondence between two ways of decomposition of curves induced by cages.

- Robert Milewski.
Upper and Lower Sequence on the Cage. Part II,
Formalized Mathematics 9(4), pages 817-823, 2001. MML Identifier: JORDAN1G

**Summary**:

- Tetsuya Tsunetou, Grzegorz Bancerek, Yatsuka Nakamura.
Zero-Based Finite Sequences,
Formalized Mathematics 9(4), pages 825-829, 2001. MML Identifier: AFINSQ_1

**Summary**:

- Andrzej Trybulec.
More on the External Approximation of a~Continuum,
Formalized Mathematics 9(4), pages 831-841, 2001. MML Identifier: JORDAN1H

**Summary**: The main goal was to prove two facts: \begin{itemize} \itemsep-3pt \item the gauge is the Go-Board of a corresponding cage, \item the left components of the complement of the curve determined by a cage are monotonic wrt the index of the approximation. \end{itemize} Some auxiliary facts are proved, too. At the end the new notion needed for the internal approximation are defined and some useful lemmas are proved.

- Andrzej Trybulec.
More on the Finite Sequences on the Plane,
Formalized Mathematics 9(4), pages 843-847, 2001. MML Identifier: TOPREAL8

**Summary**: We continue proving lemmas needed for the proof of the Jordan curve theorem. The main goal was to prove the last theorem being a mutation of the first theorem in \cite{GOBOARD3.ABS}.

- Christoph Schwarzweller.
More on Multivariate Polynomials: Monomials and Constant Polynomials,
Formalized Mathematics 9(4), pages 849-855, 2001. MML Identifier: POLYNOM7

**Summary**: In this article we give some technical concepts for multivariate polynomials with arbitrary number of variables. Monomials and constant polynomials are introduced and their properties with respect to the eval functor are shown. In addition, the multiplication of polynomials with coefficients is defined and investigated.

- Hisayoshi Kunimune, Grzegorz Bancerek, Yatsuka Nakamura.
On State Machines of Calculating Type,
Formalized Mathematics 9(4), pages 857-864, 2001. MML Identifier: FSM_2

**Summary**: In this article, we show the properties of the calculating type state machines. In the first section, we have defined calculating type state machines of which the state transition only depends on the first input. We have also proved theorems of the state machines. In the second section, we defined Moore machines with final states. We also introduced the concept of result of the Moore machines. In the last section, we proved the correctness of several calculating type of Moore machines.

- Mariusz Giero.
Hierarchies and Classifications of Sets,
Formalized Mathematics 9(4), pages 865-869, 2001. MML Identifier: TAXONOM2

**Summary**: This article is a continuation of \cite{TAXONOM1.ABS} article. Further properties of classification of sets are proved. The notion of hierarchy of a set is introduced. Properties of partitions and hierarchies are shown. The main theorem says that for each hierarchy there exists a classification which union equals to the considered hierarchy.

- Artur Kornilowicz.
On the Instructions of \SCM,
Formalized Mathematics 9(4), pages 659-663, 2001. MML Identifier: AMI_6
**Volume 10, 2002**- Number 1
- Yatsuka Nakamura.
Fan Homeomorphisms in the Plane,
Formalized Mathematics 10(1), pages 1-19, 2002. MML Identifier: JGRAPH_4

**Summary**: We will introduce four homeomorphisms (Fan morphisms) which give spoke-like distortion to the plane. They do not change the norms of vectors and preserve halfplanes invariant. These morphisms are used to regulate placement of points on the circle.

- Yatsuka Nakamura.
Half Open Intervals in Real Numbers,
Formalized Mathematics 10(1), pages 21-22, 2002. MML Identifier: RCOMP_2

**Summary**: Left and right half open intervals in the real line are defined. Their properties are investigated. A class of all finite union of such intervals are, in a sense, closed by operations of union, intersection and the difference of sets.

- Adam Naumowicz, Robert Milewski.
Some Remarks on Clockwise Oriented Sequences on Go-boards,
Formalized Mathematics 10(1), pages 23-27, 2002. MML Identifier: JORDAN1I

**Summary**: The main goal of this paper is to present alternative characterizations of clockwise oriented sequences on Go-boards.

- Gilbert Lee, Piotr Rudnicki.
Dickson's Lemma,
Formalized Mathematics 10(1), pages 29-37, 2002. MML Identifier: DICKSON

**Summary**: We present a Mizar formalization of the proof of Dickson's lemma following \cite{Becker93}, chapters 4.2 and 4.3.

- Gilbert Lee, Piotr Rudnicki.
On Ordering of Bags,
Formalized Mathematics 10(1), pages 39-46, 2002. MML Identifier: BAGORDER

**Summary**: We present a Mizar formalization of chapter 4.4 of \cite{Becker93} devoted to special orderings in additive monoids to be used for ordering terms in multivariate polynomials. We have extended the treatment to the case of infinite number of variables. It turns out that in such case admissible orderings are not necessarily well orderings.

- Grzegorz Bancerek, Shin'nosuke Yamaguchi, Yasunari Shidama.
Combining of Multi Cell Circuits,
Formalized Mathematics 10(1), pages 47-64, 2002. MML Identifier: CIRCCMB2

**Summary**: In this article we continue the investigations from \cite{CIRCCOMB.ABS} and \cite{FACIRC_1.ABS} of verification of a circuit design. We concentrate on the combination of multi cell circuits from given cells (circuit modules). Namely, we formalize a design of the form \\ \input CIRCCMB2.PIC and prove its stability. The formalization proposed consists in a series of schemes which allow to define multi cells circuits and prove their properties. Our goal is to achive mathematical formalization which will allow to verify designs of real circuits.

- Grzegorz Bancerek, Shin'nosuke Yamaguchi, Katsumi Wasaki.
Full Adder Circuit. Part II,
Formalized Mathematics 10(1), pages 65-71, 2002. MML Identifier: FACIRC_2

**Summary**: In this article we continue the investigations from \cite{FACIRC_1.ABS} of verification of a design of adder circuit. We define it as a combination of 1-bit adders using schemes from \cite{CIRCCMB2.ABS}. $n$-bit adder circuit has the following structure\\ \input FACIRC_2.PIC As the main result we prove the stability of the circuit. Further works will consist of the proof of full correctness of the circuit.

- Yatsuka Nakamura.
Fan Homeomorphisms in the Plane,
Formalized Mathematics 10(1), pages 1-19, 2002. MML Identifier: JGRAPH_4
- Number 2
- Robert Milewski.
Upper and Lower Sequence on the Cage, Upper and Lower Arcs,
Formalized Mathematics 10(2), pages 73-80, 2002. MML Identifier: JORDAN1J

**Summary**:

- Robert M. Solovay.
Fibonacci Numbers,
Formalized Mathematics 10(2), pages 81-83, 2002. MML Identifier: FIB_NUM

**Summary**: We show that Fibonacci commutes with g.c.d.; we then derive the formula connecting the Fibonacci sequence with the roots of the polynomial $x^2 - x - 1.$

- Andrzej Trybulec.
Preparing the Internal Approximations of Simple Closed Curves,
Formalized Mathematics 10(2), pages 85-87, 2002. MML Identifier: JORDAN11

**Summary**: We mean by an internal approximation of a simple closed curve a special polygon disjoint with it but sufficiently close to it, i.e. such that it is clock-wise oriented and its right cells meet the curve. We prove lemmas used in the next article to construct a sequence of internal approximations.

- Mariusz Giero.
On the General Position of Special Polygons,
Formalized Mathematics 10(2), pages 89-95, 2002. MML Identifier: JORDAN12

**Summary**: In this paper we introduce the notion of general position. We also show some auxiliary theorems for proving Jordan curve theorem. The following main theorems are proved: \begin{enumerate} \item End points of a polygon are in the same component of a complement of another polygon if number of common points of these polygons is even; \item Two points of polygon $L$ are in the same component of a complement of polygon $M$ if two points of polygon $M$ are in the same component of polygon $L.$ \end{enumerate}

- Andrzej Trybulec.
Introducing Spans,
Formalized Mathematics 10(2), pages 97-98, 2002. MML Identifier: JORDAN13

**Summary**: A sequence of internal approximations of simple closed curves is introduced. They are called spans.

- Yatsuka Nakamura.
General Fashoda Meet Theorem for Unit Circle,
Formalized Mathematics 10(2), pages 99-109, 2002. MML Identifier: JGRAPH_5

**Summary**: Outside and inside Fashoda theorems are proven for points in general position on unit circle. Four points must be ordered in a sense of ordering for simple closed curve. For preparation of proof, the relation between the order and condition of coordinates of points on unit circle is discussed.

- Robert Milewski.
Properties of the Internal Approximation of Jordan's Curve,
Formalized Mathematics 10(2), pages 111-115, 2002. MML Identifier: JORDAN14

**Summary**:

- Robert Milewski.
Upper and Lower Sequence on the Cage, Upper and Lower Arcs,
Formalized Mathematics 10(2), pages 73-80, 2002. MML Identifier: JORDAN1J
- Number 3
- Grzegorz Bancerek, Adam Naumowicz.
Preliminaries to Automatic Generation of Mizar Documentation for Circuits,
Formalized Mathematics 10(3), pages 117-133, 2002. MML Identifier: CIRCCMB3

**Summary**: In this paper we introduce technical notions used by a system which automatically generates Mizar documentation for specified circuits. They provide a ready for use elements needed to justify correctness of circuits' construction. We concentrate on the concept of stabilization and analyze one-gate circuits and their combinations.

- Robert Milewski.
Properties of the Upper and Lower Sequence on the Cage,
Formalized Mathematics 10(3), pages 135-143, 2002. MML Identifier: JORDAN15

**Summary**:

- Adam Grabowski.
On the Decompositions of Intervals and Simple Closed Curves,
Formalized Mathematics 10(3), pages 145-151, 2002. MML Identifier: BORSUK_4

**Summary**: The aim of the paper is to show that the only subcontinua of the Jordan curve are arcs, the whole curve, and singletons of its points. Additionally, it has been shown that the only subcontinua of the unit interval $\Bbb I$ are closed intervals.

- Andrzej Trybulec.
On the Minimal Distance Between Sets in Euclidean Space,
Formalized Mathematics 10(3), pages 153-158, 2002. MML Identifier: JORDAN1K

**Summary**: The concept of the minimal distance between two sets in a Euclidean space is introduced and some useful lemmas are proved.

- Yatsuka Nakamura, Andrzej Trybulec.
Sequences of Metric Spaces and an Abstract Intermediate Value Theorem,
Formalized Mathematics 10(3), pages 159-161, 2002. MML Identifier: TOPMETR3

**Summary**: Relations of convergence of real sequences and convergence of metric spaces are investigated. An abstract intermediate value theorem for two closed sets in the range is presented. At the end, it is proven that an arc connecting the west minimal point and the east maximal point in a simple closed curve must be identical to the upper arc or lower arc of the closed curve.

- Andrzej Trybulec, Yatsuka Nakamura.
On the Decomposition of a Simple Closed Curve into Two Arcs,
Formalized Mathematics 10(3), pages 163-167, 2002. MML Identifier: JORDAN16

**Summary**: The purpose of the paper is to prove lemmas needed for the Jordan curve theorem. The main result is that the decomposition of a simple closed curve into two arcs with the ends $p_1, p_2$ is unique in the sense that every arc on the curve with the same ends must be equal to one of them.

- Artur Kornilowicz.
The Ordering of Points on a Curve. Part III,
Formalized Mathematics 10(3), pages 169-171, 2002. MML Identifier: JORDAN17

**Summary**:

- Artur Kornilowicz.
The Ordering of Points on a Curve. Part IV,
Formalized Mathematics 10(3), pages 173-177, 2002. MML Identifier: JORDAN18

**Summary**:

- Josef Urban.
Order Sorted Algebras,
Formalized Mathematics 10(3), pages 179-188, 2002. MML Identifier: OSALG_1

**Summary**: Initial notions for order sorted algebras.

- Josef Urban.
Subalgebras of an Order Sorted Algebra. Lattice of Subalgebras,
Formalized Mathematics 10(3), pages 189-196, 2002. MML Identifier: OSALG_2

**Summary**:

- Josef Urban.
Homomorphisms of Order Sorted Algebras,
Formalized Mathematics 10(3), pages 197-200, 2002. MML Identifier: OSALG_3

**Summary**:

- Josef Urban.
Order Sorted Quotient Algebra,
Formalized Mathematics 10(3), pages 201-210, 2002. MML Identifier: OSALG_4

**Summary**:

- Josef Urban.
Free Order Sorted Universal Algebra,
Formalized Mathematics 10(3), pages 211-225, 2002. MML Identifier: OSAFREE

**Summary**: Free Order Sorted Universal Algebra --- the general construction for any locally directed signatures.

- Grzegorz Bancerek, Adam Naumowicz.
Preliminaries to Automatic Generation of Mizar Documentation for Circuits,
Formalized Mathematics 10(3), pages 117-133, 2002. MML Identifier: CIRCCMB3
**Volume 11, 2003**- Number 1
- Noboru Endou, Takashi Mitsuishi, Yasunari Shidama.
Subspaces and Cosets of Subspace of Real Unitary Space,
Formalized Mathematics 11(1), pages 1-7, 2003. MML Identifier: RUSUB_1

**Summary**: In this article, subspace and the coset of subspace of real unitary space are defined. And we discuss some of their fundamental properties.

- Noboru Endou, Takashi Mitsuishi, Yasunari Shidama.
Operations on Subspaces in Real Unitary Space,
Formalized Mathematics 11(1), pages 9-16, 2003. MML Identifier: RUSUB_2

**Summary**: In this article, we extend an operation of real linear space to real unitary space. We show theorems proved in \cite{RLSUB_2.ABS} on real unitary space.

- Noboru Endou, Takashi Mitsuishi, Yasunari Shidama.
Linear Combinations in Real Unitary Space,
Formalized Mathematics 11(1), pages 17-22, 2003. MML Identifier: RUSUB_3

**Summary**: In this article, we mainly discuss linear combination of vectors in Real Unitary Space and dimension of the space. As the result, we obtain some theorems that are similar to those in Real Linear Space.

- Noboru Endou, Takashi Mitsuishi, Yasunari Shidama.
Dimension of Real Unitary Space,
Formalized Mathematics 11(1), pages 23-28, 2003. MML Identifier: RUSUB_4

**Summary**: In this article we describe the dimension of real unitary space. Most of theorems are restricted from real linear space. In the last section, we introduce affine subset of real unitary space.

- Takashi Mitsuishi, Noboru Endou, Keiji Ohkubo.
Trigonometric Functions on Complex Space,
Formalized Mathematics 11(1), pages 29-32, 2003. MML Identifier: SIN_COS3

**Summary**: This article describes definitions of sine, cosine, hyperbolic sine and hyperbolic cosine. Some of their basic properties are discussed.

- Noboru Endou, Takashi Mitsuishi, Yasunari Shidama.
Topology of Real Unitary Space,
Formalized Mathematics 11(1), pages 33-38, 2003. MML Identifier: RUSUB_5

**Summary**: In this article we introduce three subjects in real unitary space: parallelism of subsets, orthogonality of subsets and topology of the space. In particular, to introduce the topology of real unitary space, we discuss the metric topology which is induced by the inner product in the space. As the result, we are able to discuss some topological subjects on real unitary space.

- William W. Armstrong, Yatsuka Nakamura, Piotr Rudnicki.
Armstrong's Axioms,
Formalized Mathematics 11(1), pages 39-51, 2003. MML Identifier: ARMSTRNG

**Summary**: We present a formalization of the seminal paper by W.~W.~Armstrong~\cite{arm74} on functional dependencies in relational data bases. The paper is formalized in its entirety including examples and applications. The formalization was done with a routine effort albeit some new notions were defined which simplified formulation of some theorems and proofs.\par The definitive reference to the theory of relational databases is~\cite{Maier}, where saturated sets are called closed sets. Armstrong's ``axioms'' for functional dependencies are still widely taught at all levels of database design, see for instance~\cite{Elmasri}.

- Noboru Endou, Takashi Mitsuishi, Yasunari Shidama.
Convex Sets and Convex Combinations,
Formalized Mathematics 11(1), pages 53-58, 2003. MML Identifier: CONVEX1

**Summary**: Convexity is one of the most important concepts in a study of analysis. Especially, it has been applied around the optimization problem widely. Our purpose is to define the concept of convexity of a set on Mizar, and to develop the generalities of convex analysis. The construction of this article is as follows: Convexity of the set is defined in the section 1. The section 2 gives the definition of convex combination which is a kind of the linear combination and related theorems are proved there. In section 3, we define the convex hull which is an intersection of all convex sets including a given set. The last section is some theorems which are necessary to compose this article.

- Jaroslaw Kotowicz.
Quotient Vector Spaces and Functionals,
Formalized Mathematics 11(1), pages 59-68, 2003. MML Identifier: VECTSP10

**Summary**: The article presents well known facts about quotient vector spaces and functionals (see \cite{SLang}). There are repeated theorems and constructions with either weaker assumptions or in more general situations (see \cite{HAHNBAN.ABS}, \cite{VECTSP_1.ABS}, \cite{LMOD_7.ABS}). The construction of coefficient functionals and non degenerated functional in quotient vector space generated by functional in the given vector space are the only new things which are done.

- Jaroslaw Kotowicz.
Bilinear Functionals in Vector Spaces,
Formalized Mathematics 11(1), pages 69-86, 2003. MML Identifier: BILINEAR

**Summary**: The main goal of the article is the presentation of the theory of bilinear functionals in vector spaces. It introduces standard operations on bilinear functionals and proves their classical properties. It is shown that quotient functionals are non degenerated on the left and the right. In the case of symmetric and alternating bilinear functionals it is shown that the left and right kernels are equal.

- Jaroslaw Kotowicz.
Hermitan Functionals. Canonical Construction of Scalar Product in Quotient Vector Space,
Formalized Mathematics 11(1), pages 87-98, 2003. MML Identifier: HERMITAN

**Summary**: In the article we present antilinear functionals, sesquilinear and hermitan forms. We prove Schwarz and Minkowski inequalities, and Parallelogram Law for non negative hermitan form. The proof of Schwarz inequality is based on \cite{RUDIN:2}. The incorrect proof of this fact can be found in \cite{Maurin}. The construction of scalar product in quotient vector space from non negative hermitan functions is the main result of the article.

- Krzysztof Retel.
The Class of Series -- Parallel Graphs. Part I,
Formalized Mathematics 11(1), pages 99-103, 2003. MML Identifier: NECKLACE

**Summary**: The paper introduces some preliminary notions concerning the graph theory according to \cite{Thomasse}. We define Necklace $n$ as a graph with vertex $\{1,2,3,\dots,n\}$ and edge set $\{(1,2),(2,3),\dots,(n-1,n)\}.$ The goal of the article is to prove that Necklace $n$ and Complement of Necklace $n$ are isomorphic for $n = 0, 1, 4.$

- Christoph Schwarzweller.
Term Orders,
Formalized Mathematics 11(1), pages 105-111, 2003. MML Identifier: TERMORD

**Summary**: We continue the formalization of \cite{Becker93} towards Gr\"obner Bases. Here we deal with term orders, that is with orders on bags. We introduce the notions of head term, head coefficient, and head monomial necessary to define reduction for polynomials.

- Christoph Schwarzweller.
Polynomial Reduction,
Formalized Mathematics 11(1), pages 113-123, 2003. MML Identifier: POLYRED

**Summary**: We continue the formalization of \cite{Becker93} towards Gr\"obner Bases. In this article we introduce reduction of polynomials and prove its termination, its adequateness for ideal congruence as well as the translation lemma used later to show confluence of reduction.

- Grzegorz Bancerek, Mitsuru Aoki, Akio Matsumoto, Yasunari Shidama.
Processes in Petri nets,
Formalized Mathematics 11(1), pages 125-132, 2003. MML Identifier: PNPROC_1

**Summary**: Sequential and concurrent compositions of processes in Petri nets are introduced. A process is formalized as a set of (possible), so called, firing sequences. In the definition of the sequential composition the standard concatenation is used $$ R_1 \mathop{\rm before} R_2 = \{p_1\mathop{^\frown}p_2: p_1\in R_1\ \land\ p_2\in R_2\} $$ The definition of the concurrent composition is more complicated $$ R_1 \mathop{\rm concur} R_2 = \{ q_1\cup q_2: q_1\ {\rm misses}\ q_2\ \land\ \mathop{\rm Seq} q_1\in R_1\ \land\ \mathop{\rm Seq} q_2\in R_2\} $$ For example, $$ \{\langle t_0\rangle\} \mathop{\rm concur} \{\langle t_1,t_2\rangle\} = \{\langle t_0,t_1,t_2\rangle , \langle t_1,t_0,t_2\rangle , \langle t_1,t_2,t_0\rangle\} $$ The basic properties of the compositions are shown.

- Noboru Endou, Takashi Mitsuishi, Yasunari Shidama.
Subspaces and Cosets of Subspace of Real Unitary Space,
Formalized Mathematics 11(1), pages 1-7, 2003. MML Identifier: RUSUB_1
- Number 2
- Masaaki Niimura, Yasushi Fuwa.
Improvement of Radix-$2^k$ Signed-Digit Number for High Speed Circuit,
Formalized Mathematics 11(2), pages 133-137, 2003. MML Identifier: RADIX_3

**Summary**: In this article, a new radix-$2^k$ signed-digit number (Radix-$2^k$ sub signed-digit number) is defined and its properties for hardware realization are discussed. \par Until now, high speed calculation method with Radix-$2^k$ signed-digit numbers is proposed, but this method used ``Compares With 2" to calculate carry. ``Compares with 2'' is a very simple method, but it needs very complicated hardware especially when the value of $k$ becomes large. In this article, we propose a subset of Radix-$2^k$ signed-digit, named Radix-$2^k$ sub signed-digit numbers. Radix-$2^k$ sub signed-digit was designed so that the carry calculation use ``bit compare'' to hardware-realization simplifies more.\par In the first section of this article, we defined the concept of Radix-$2^k$ sub signed-digit numbers and proved some of their properties. In the second section, we defined the new carry calculation method in consideration of hardware-realization, and proved some of their properties. In the third section, we provide some functions for generating Radix-$2^k$ sub signed-digit numbers from Radix-$2^k$ signed-digit numbers. In the last section, we defined some functions for generation natural numbers from Radix-$2^k$ sub signed-digit, and we clarified its correctness.

- Masaaki Niimura, Yasushi Fuwa.
High Speed Adder Algorithm with Radix-$2^k$ Sub Signed-Digit Number,
Formalized Mathematics 11(2), pages 139-141, 2003. MML Identifier: RADIX_4

**Summary**: In this article, a new adder algorithm using Radix-$2^k$ sub signed-digit numbers is defined and properties for the hardware-realization is discussed.\par Until now, we proposed Radix-$2^k$ sub signed-digit numbers in consideration of the hardware realization. In this article, we proposed High Speed Adder Algorithm using this Radix-$2^k$ sub signed-digit numbers. This method has two ways to speed up at hardware-realization. One is 'bit compare' at carry calculation, it is proposed in another article. Other is carry calculation between two numbers. We proposed that $n$ digits Radix-$2^k$ signed-digit numbers is expressed in $n+1$ digits Radix-$2^k$ sub signed-digit numbers, and addition result of two $n+1$ digits Radix-$2^k$ sub signed-digit numbers is expressed in $n+1$ digits. In this way, carry operation between two Radix-$2^k$ sub signed-digit numbers can be processed at $n+1$ digit adder circuit and additional circuit to operate carry is not needed.\par In the first section of this article, we prepared some useful theorems for operation of Radix-$2^k$ numbers. In the second section, we proved some properties about carry on Radix-$2^k$ sub signed-digit numbers. In the last section, we defined the new addition operation using Radix-$2^k$ sub signed-digit numbers, and we clarified its correctness.

- Jing-Chao Chen, Yatsuka Nakamura.
The Underlying Principle of Dijkstra's Shortest Path Algorithm,
Formalized Mathematics 11(2), pages 143-152, 2003. MML Identifier: GRAPH_5

**Summary**: A path from a source vertex $v$ to a target vertex $u$ is said to be a shortest path if its total cost is minimum among all $v$-to-$u$ paths. Dijkstra's algorithm is a classic shortest path algorithm, which is described in many textbooks. To justify its correctness (whose rigorous proof will be given in the next article), it is necessary to clarify its underlying principle. For this purpose, the article justifies the following basic facts, which are the core of Dijkstra's algorithm. \begin{itemize} \itemsep-3pt \item A graph is given, its vertex set is denoted by $V.$ Assume $U$ is the subset of $V,$ and if a path $p$ from $s$ to $t$ is the shortest among the set of paths, each of which passes through only the vertices in $U,$ except the source and sink, and its source and sink is $s$ and in $V,$ respectively, then $p$ is a shortest path from $s$ to $t$ in the graph, and for any subgraph which contains at least $U,$ it is also the shortest. \item Let $p(s,x,U)$ denote the shortest path from $s$ to $x$ in a subgraph whose the vertex set is the union of $\{s,x\}$ and $U,$ and cost $(p)$ denote the cost of path $p(s,x,U),$ cost$(x,y)$ the cost of the edge from $x$ to $y.$ Give $p(s,x,U),$ $q(s,y,U)$ and $r(s,y,U \cup \{x\})$. If ${\rm cost}(p) = {\rm min} \{{\rm cost}(w): w(s,t,U) \wedge t \in V\}$, then we have $${\rm cost}(r) = {\rm min} ({\rm cost}(p)+{\rm cost}(x,y),{\rm cost}(q)).$$ \end{itemize} \noindent This is the well-known triangle comparison of Dijkstra's algorithm.

- Adam Grabowski.
On the Hausdorff Distance Between Compact Subsets,
Formalized Mathematics 11(2), pages 153-157, 2003. MML Identifier: HAUSDORF

**Summary**: In \cite{WEIERSTR.ABS} the pseudo-metric ${\rm dist}^{\rm max}_{\rm min}$ on compact subsets $A$ and $B$ of a topological space generated from arbitrary metric space is defined. Using this notion we define the Hausdorff distance (see e.g. \cite{Csaszar}) of $A$ and $B$ as a maximum of the two pseudo-distances: from $A$ to $B$ and from $B$ to $A$. We justify its distance properties. At the end we define some special notions which enable to apply the Hausdorff distance operator ${\rm ``HausDist"}$ to the subsets of the Euclidean topological space~$\calE^n_T.$

- Freek Wiedijk.
Chains on a Grating in Euclidean Space,
Formalized Mathematics 11(2), pages 159-167, 2003. MML Identifier: CHAIN_1

**Summary**: Translation of pages 101, the second half of 102, and 103 of \cite{Newman51}.

- Hiroshi Yamazaki, Yasunari Shidama, Yatsuka Nakamura.
Bessel's Inequality,
Formalized Mathematics 11(2), pages 169-173, 2003. MML Identifier: BHSP_5

**Summary**: In this article we defined the operation of a set and proved Bessel's inequality. In the first section, we defined the sum of all results of an operation, in which the results are given by taking each element of a set. In the second section, we defined Orthogonal Family and Orthonormal Family. In the last section, we proved some properties of operation of set and Bessel's inequality.

- Hisayoshi Kunimune, Yatsuka Nakamura.
A Representation of Integers by Binary Arithmetics and Addition of Integers,
Formalized Mathematics 11(2), pages 175-178, 2003. MML Identifier: BINARI_4

**Summary**: In this article, we introduce the new concept of 2's complement representation. Natural numbers that are congruent mod $n$ can be represented by the same $n$ bits binary. Using the concept introduced here, negative numbers that are congruent mod $n$ also can be represented by the same $n$ bit binary. We also show some properties of addition of integers using this concept.

- Kanchun , Yatsuka Nakamura.
The Inner Product of Finite Sequences and of Points of $n$-dimensional Topological Space,
Formalized Mathematics 11(2), pages 179-183, 2003. MML Identifier: EUCLID_2

**Summary**: First, we define the inner product to finite sequences of real value. Next, we extend it to points of $n$-dimensional topological space ${\calE}^{n}_{\rmT}$. At the end, orthogonality is introduced to this space.

- Xiquan Liang.
Solving Roots of Polynomial Equation of Degree 4 with Real Coefficients,
Formalized Mathematics 11(2), pages 185-187, 2003. MML Identifier: POLYEQ_2

**Summary**: In this paper, we describe the definition of the fourth degree algebraic equations and their properties. We clarify the relation between the four roots of this equation and its coefficient. Also, the form of these roots for various conditions is discussed. This solution is known as the Cardano solution.

- Artur Kornilowicz.
Morphisms Into Chains. Part I,
Formalized Mathematics 11(2), pages 189-195, 2003. MML Identifier: WAYBEL35

**Summary**: This work is the continuation of formalization of \cite{CCL}. Items from 2.1 to 2.8 of Chapter 4 are proved.

- Shunichi Kobayashi.
Propositional Calculus for Boolean Valued Functions. Part VII,
Formalized Mathematics 11(2), pages 197-199, 2003. MML Identifier: BVFUNC25

**Summary**: In this paper, we proved some elementary propositional calculus formulae for Boolean valued functions.

- Markus Moschner.
Basic Notions and Properties of Orthoposets,
Formalized Mathematics 11(2), pages 201-210, 2003. MML Identifier: OPOSET_1

**Summary**: Orthoposets are defined. The approach is the standard one via order relation similar to common text books on algebra like \cite{Gratzer}.

- Masaaki Niimura, Yasushi Fuwa.
Improvement of Radix-$2^k$ Signed-Digit Number for High Speed Circuit,
Formalized Mathematics 11(2), pages 133-137, 2003. MML Identifier: RADIX_3
- Number 3
- Yatsuka Nakamura.
General Fashoda Meet Theorem for Unit Circle and Square,
Formalized Mathematics 11(3), pages 213-224, 2003. MML Identifier: JGRAPH_6

**Summary**: Here we will prove Fashoda meet theorem for the unit circle and for a square, when 4 points on the boundary are ordered cyclically. Also, the concepts of general rectangle and general circle are defined.

- Hiroshi Yamazaki, Yasumasa Suzuki, Takao Inoue, Yasunari Shidama.
On Some Properties of Real Hilbert Space. Part I,
Formalized Mathematics 11(3), pages 225-229, 2003. MML Identifier: BHSP_6

**Summary**: In this paper, we first introduce the notion of summability of an infinite set of vectors of real Hilbert space, without using index sets. Further we introduce the notion of weak summability, which is weaker than that of summability. Then, several statements for summable sets and weakly summable ones are proved. In the last part of the paper, we give a necessary and sufficient condition for summability of an infinite set of vectors of real Hilbert space as our main theorem. The last theorem is due to \cite{Halmos87}.

- Shin'nosuke Yamaguchi, Grzegorz Bancerek, Katsumi Wasaki.
Full Subtracter Circuit. Part II,
Formalized Mathematics 11(3), pages 231-236, 2003. MML Identifier: FSCIRC_2

**Summary**: In this article we continue investigations from \cite{FSCIRC_1.ABS} of verification of a design of subtracter circuit. We define it as a combination of multi cell circuit using schemes from \cite{CIRCCMB2.ABS}. As the main result we prove the stability of the circuit.

- Jing-Chao Chen.
Dijkstra's Shortest Path Algorithm,
Formalized Mathematics 11(3), pages 237-247, 2003. MML Identifier: GRAPHSP

**Summary**: The article formalizes Dijkstra's shortest path algorithm \cite{Dijkstra59}. A path from a source vertex $v$ to $a$ target vertex $u$ is said to be the shortest path if its total cost is minimum among all $v$-to-$u$ paths. Dijkstra's algorithm is based on the following assumptions: \begin{itemize} \item All edge costs are non-negative. \item The number of vertices is finite. \item The source is a single vertex, but the target may be all other vertices. \end{itemize} The underlying principle of the algorithm may be described as follows: the algorithm starts with the source; it visits the vertices in order of increasing cost, and maintains a set $V$ of visited vertices (denoted by UsedVx in the article) whose cost from the source has been computed, and a tentative cost $D(u)$ to each unvisited vertex $u.$ In the article, the set of all unvisited vertices is denoted by UnusedVx. $D(u)$ is the cost of the shortest path from the source to u in the subgraph induced by $V \cup \{u\}.$ We denote the set of all unvisited vertices whose $D$-values are not infinite (i.e. in the subgraph each of which has a path from the source to itself) by OuterVx. Dijkstra's algorithm repeatedly searches OuterVx for the vertex with minimum tentative cost (this procedure is called findmin in the article), adds it to the set $V$ and modifies $D$-values by a procedure, called Relax. Suppose the unvisited vertex with minimum tentative cost is $x$, the procedure Relax replaces $D(u)$ with min$\{D(u),D(u)+cost(x,u)\}$ where $u$ is a vertex in UnusedVx, and cost$(x,u)$ is the cost of edge $(x,u).$ In the Mizar library, there are several computer models, e.g. SCMFSA and SCMPDS etc. However, it is extremely difficult to use these models to formalize the algorithm. Instead, we adopt functions in the Mizar library, which seem to be pseudo-codes, and are similar to those in the functional programming language, e.g. Lisp. To date, there is no rigorous justification with respect to the correctness of Dijkstra's algorithm. The article presents first the rigorous justification.

- Noboru Endou, Yasumasa Suzuki, Yasunari Shidama.
Real Linear Space of Real Sequences,
Formalized Mathematics 11(3), pages 249-253, 2003. MML Identifier: RSSPACE

**Summary**: The article is a continuation of \cite{RLVECT_1.ABS}. As the example of real linear spaces, we introduce the arithmetic addition in the set of real sequences and also introduce the multiplication. This set has the arithmetic structure which depends on these arithmetic operations.

- Noboru Endou, Yasumasa Suzuki, Yasunari Shidama.
Hilbert Space of Real Sequences,
Formalized Mathematics 11(3), pages 255-257, 2003. MML Identifier: RSSPACE2

**Summary**: A continuation of \cite{RLVECT_1.ABS}. As the example of real unitary spaces, we introduce the arithmetic addition and multiplication in the set of square sum able real sequences and introduce the scaler products also. This set has the structure of the Hilbert space.

- Takao Inoue.
Intuitionistic Propositional Calculus in the Extended Framework with Modal Operator. Part I,
Formalized Mathematics 11(3), pages 259-266, 2003. MML Identifier: INTPRO_1

**Summary**: In this paper, we develop intuitionistic propositional calculus IPC in the extended language with single modal operator. The formulation that we adopt in this paper is very useful not only to formalize the calculus but also to do a number of logics with essentially propositional character. In addition, it is much simpler than the past formalization for modal logic. In the first section, we give the mentioned formulation which the author heavily owes to the formalism of Adam Grabowski's \cite{HILBERT1.ABS}. After the theoretical development of the logic, we prove a number of valid formulas of IPC in the sections 2--4. The last two sections are devoted to present classical propositional calculus and modal calculus S4 in our framework, as a preparation for future study. In the forthcoming Part II of this paper, we shall prove, among others, a number of intuitionistically valid formulas with negation.

- Noboru Endou, Yasumasa Suzuki, Yasunari Shidama.
Some Properties for Convex Combinations,
Formalized Mathematics 11(3), pages 267-270, 2003. MML Identifier: CONVEX2

**Summary**: This is a continuation of \cite{CONVEX1.ABS}. In this article, we proved that convex combination on convex family is convex.

- Hiroshi Yamazaki, Yasumasa Suzuki, Takao Inoue, Yasunari Shidama.
On Some Properties of Real Hilbert Space. Part II,
Formalized Mathematics 11(3), pages 271-273, 2003. MML Identifier: BHSP_7

**Summary**: This paper is a continuation of our paper \cite{BHSP_6.ABS}. We give an analogue of the necessary and sufficient condition for summable set (i.e. the main theorem of \cite{BHSP_6.ABS}) with respect to summable set by a functional $L$ in real Hilbert space. After presenting certain useful lemmas, we prove our main theorem that the summability for an orthonormal infinite set in real Hilbert space is equivalent to its summability with respect to the square of norm, say $H(x) = (x, x)$. Then we show that the square of norm $H$ commutes with infinite sum operation if the orthonormal set under our consideration is summable. Our main theorem is due to \cite{Halmos87}.

- Wenpai Chang, Yatsuka Nakamura, Piotr Rudnicki.
Inner Products and Angles of Complex Numbers,
Formalized Mathematics 11(3), pages 275-280, 2003. MML Identifier: COMPLEX2

**Summary**: An inner product of complex numbers is defined and used to characterize the (counter-clockwise) angle between ($a$,0) and (0,$b$) in the complex plane. For complex $a$, $b$ and $c$ we then define the (counter-clockwise) angle between ($a$,$c$) and ($c$, $b$) and prove theorems about the sum of internal and external angles of a triangle.

- Akihiro Kubo, Yatsuka Nakamura.
Angle and Triangle in Euclidian Topological Space,
Formalized Mathematics 11(3), pages 281-287, 2003. MML Identifier: EUCLID_3

**Summary**: Two transformations between the complex space and 2-dimensional Euclidian topological space are defined. By them, the concept of argument is induced to 2-dimensional vectors using argument of complex number. Similarly, the concept of an angle is introduced using the angle of two complex numbers. The concept of a triangle and related concepts are also defined in $n$-dimensional Euclidian topological spaces.

- Krzysztof Retel.
The Class of Series-Parallel Graphs. Part II,
Formalized Mathematics 11(3), pages 289-291, 2003. MML Identifier: NECKLA_2

**Summary**: In this paper we introduce two new operations on graphs: sum and union corresponding to parallel and series operation respectively. We determine $N$-free graph as the graph that does not embed Necklace $4$. We define ``fin\_RelStr" as the set of all graphs with finite carriers. We also define the smallest class of graphs which contains the one-element graph and which is closed under parallel and series operations. The goal of the article is to prove the theorem that the class of finite series-parallel graphs is the class of finite $N$-free graphs. This paper formalizes the next part of \cite{Thomasse}.

- Christoph Schwarzweller.
Characterization and Existence of Gr\"obner Bases,
Formalized Mathematics 11(3), pages 293-301, 2003. MML Identifier: GROEB_1

**Summary**: We continue the Mizar formalization of Gr\"{o}bner bases following \cite{Becker93}. In this article we prove a number of characterizations of Gr\"{o}bner bases among them that Gr\"{o}bner bases are convergent rewriting systems. We also show the existence and uniqueness of reduced Gr\"{o}bner bases.

- Christoph Schwarzweller.
Construction of Gr\"obner bases. S-Polynomials and Standard Representations,
Formalized Mathematics 11(3), pages 303-312, 2003. MML Identifier: GROEB_2

**Summary**: We continue the Mizar formalization of Gr\"{o}bner bases following \cite{Becker93}. In this article we introduce S-polynomials and standard representations and show how these notions can be used to characterize Gr\"{o}bner bases.

- Adam Grabowski.
On the Subcontinua of a Real Line,
Formalized Mathematics 11(3), pages 313-322, 2003. MML Identifier: BORSUK_5

**Summary**: In \cite{BORSUK_4.ABS} we showed that the only proper subcontinua of the simple closed curve are arcs and single points. In this article we prove that the only proper subcontinua of the real line are closed intervals. We introduce some auxiliary notions such as $\rbrack a,b\lbrack_{\Bbb Q}$, $\rbrack a,b\lbrack_{\Bbb I\Bbb Q}$ -- intervals consisting of rational and irrational numbers respectively. We show also some basic topological properties of intervals.

- Lilla Krystyna Baginska, Adam Grabowski.
On the Kuratowski Closure-Complement Problem,
Formalized Mathematics 11(3), pages 323-329, 2003. MML Identifier: KURATO_1

**Summary**: In this article we formalize the Kuratowski closure-complement result: there is at most 14 distinct sets that one can produce from a given subset $A$ of a topological space $T$ by applying closure and complement operators and that all 14 can be obtained from a suitable subset of $\Bbb R,$ namely KuratExSet $=\{1\} \cup {\Bbb Q} (2,3) \cup (3, 4)\cup (4,\infty)$.\par The second part of the article deals with the maximal number of distinct sets which may be obtained from a given subset $A$ of $T$ by applying closure and interior operators. The subset KuratExSet of $\Bbb R$ is also enough to show that 7 can be achieved.

- Noboru Endou, Yasunari Shidama.
Convex Hull, Set of Convex Combinations and Convex Cone,
Formalized Mathematics 11(3), pages 331-333, 2003. MML Identifier: CONVEX3

**Summary**: In this article, there are two themes. One of them is the proof that convex hull of a given subset $M$ consists of all convex combinations of $M.$ Another is definitions of cone and convex cone and some properties of them.

- Wioletta Truszkowska, Adam Grabowski.
On the Two Short Axiomatizations of Ortholattices,
Formalized Mathematics 11(3), pages 335-340, 2003. MML Identifier: ROBBINS2

**Summary**: In the paper, two short axiom systems for Boolean algebras are introduced. In the first section we show that the single axiom (DN${}_1$) proposed in \cite{McCune:2001} in terms of disjunction and negation characterizes Boolean algebras. To prove that (DN${}_1$) is a single axiom for Robbins algebras (that is, Boolean algebras as well), we use the Otter theorem prover. The second section contains proof that the two classical axioms (Meredith${}_1$), (Meredith${}_2$) proposed by Meredith \cite{Meredith:1968} may also serve as a basis for Boolean algebras. The results will be used to characterize ortholattices.

- Yatsuka Nakamura.
General Fashoda Meet Theorem for Unit Circle and Square,
Formalized Mathematics 11(3), pages 213-224, 2003. MML Identifier: JGRAPH_6
- Number 4
- Andrzej Trybulec.
On the Sets Inhabited by Numbers,
Formalized Mathematics 11(4), pages 341-347, 2003. MML Identifier: MEMBERED

**Summary**: The information that all members of a set enjoy a property expressed by an adjective can be processed in a systematic way. The purpose of the work is to find out how to do that. If it works, `membered' will become a reserved word and the work with it will be automated. I have chosen {\it membered} rather than {\it inhabited} because of the compatibility with the Automath terminology. The phrase $\tau$ {\it inhabits} $\theta$ could be translated to $\tau$ {\bfseries\itshape is} $\theta$ in Mizar.

- Grigory E. Ivanov.
Definition of Convex Function and Jensen's Inequality,
Formalized Mathematics 11(4), pages 349-354, 2003. MML Identifier: CONVFUN1

**Summary**: Convexity of a function in a real linear space is defined as convexity of its epigraph according to ``Convex analysis'' by R. Tyrrell Rockafellar. The epigraph of a function is a subset of the product of the function's domain space and the space of real numbers. Therefore the product of two real linear spaces should be defined. The values of the functions under consideration are extended real numbers. We define the sum of a finite sequence of extended real numbers and get some properties of the sum. The relation between notions ``function is convex'' and ``function is convex on set'' (see RFUNCT\_3:def 13) is established. We obtain another version of the criterion for a set to be convex (see CONVEX2:6 to compare) that may be more suitable in some cases. Finally we prove Jensen's inequality (both strict and not strict) as criteria for functions to be convex.

- Grzegorz Bancerek.
On Semilattice Structure of Mizar Types,
Formalized Mathematics 11(4), pages 355-369, 2003. MML Identifier: ABCMIZ_0

**Summary**: The aim of this paper is to develop a formal theory of Mizar types. The presented theory is an approach to the structure of Mizar types as a sup-semilattice with widening (subtyping) relation as the order. It is an abstraction from the existing implementation of the Mizar verifier and formalization of the ideas from \cite{Bancerek:2003}.

- Akihiro Kubo.
Lines in $n$-Dimensional Euclidean Spaces,
Formalized Mathematics 11(4), pages 371-376, 2003. MML Identifier: EUCLID_4

**Summary**: In this paper, we define the line of $n$-dimensional Euclidian space and we introduce basic properties of affine space on this space. Next, we define the inner product of elements of this space. At the end, we introduce orthogonality of lines of this space.

- Yasumasa Suzuki, Noboru Endou, Yasunari Shidama.
Banach Space of Absolute Summable Real Sequences,
Formalized Mathematics 11(4), pages 377-380, 2003. MML Identifier: RSSPACE3

**Summary**: A continuation of \cite{RSSPACE2.ABS}. As the example of real norm spaces, we introduce the arithmetic addition and multiplication in the set of absolute summable real sequences and introduce the norm also. This set has the structure of the Banach space.

- Kanchun , Hiroshi Yamazaki, Yatsuka Nakamura.
Cross Products and Tripple Vector Products in 3-dimensional Euclidean Space,
Formalized Mathematics 11(4), pages 381-383, 2003. MML Identifier: EUCLID_5

**Summary**: First, we extend the basic theorems of 3-dimensional euclidian space, and then define the cross product in the same space and relative vector relations using the above definition.

- Yatsuka Nakamura, Hiroshi Yamazaki.
Calculation of Matrices of Field Elements. Part I,
Formalized Mathematics 11(4), pages 385-391, 2003. MML Identifier: MATRIX_4

**Summary**: This article gives property of calculation of matrices.

- Takashi Mitsuishi, Grzegorz Bancerek.
Lattice of Fuzzy Sets,
Formalized Mathematics 11(4), pages 393-398, 2003. MML Identifier: LFUZZY_0

**Summary**: This article concerns a connection of fuzzy logic and lattice theory. Namely, the fuzzy sets form a Heyting lattice with union and intersection of fuzzy sets as meet and join operations. The lattice of fuzzy sets is defined as the product of interval posets. As the final result, we have characterized the composition of fuzzy relations in terms of lattice theory and proved its associativity.

- Adam Grabowski.
On the Kuratowski Limit Operators,
Formalized Mathematics 11(4), pages 399-409, 2003. MML Identifier: KURATO_2

**Summary**: In the paper we give formal descriptions of the two Kuratowski limit oprators: Li $S$ and Ls $S$, where $S$ is an arbitrary sequence of subsets of a fixed topological space. In the two last sections we prove basic properties of these lower and upper topological limits, which may be found e.g. in \cite{KURAT:2}. In the sections 2--4, we present three operators which are associated in some sense with the above mentioned, that is lim inf $F$, lim sup $F$, and limes $F$, where $F$ is a sequence of subsets of a fixed 1-sorted structure.

- Andrzej Trybulec.
On the Segmentation of a Simple Closed Curve,
Formalized Mathematics 11(4), pages 411-416, 2003. MML Identifier: JORDAN_A

**Summary**: The main goal of the work was to introduce the concept of the segmentation of a simple closed curve into (arbitrary small) arcs. The existence of it has been proved by Yatsuka Nakamura \cite{JORDAN7.ABS}. The concept of the gap of a segmentation is also introduced. It is the smallest distance between disjoint segments in the segmentation. For this purpose, the relationship between segments of an arc \cite{JORDAN6.ABS} and segments on a simple closed curve \cite{JORDAN7.ABS} has been shown.

- Shunichi Kobayashi.
On the Calculus of Binary Arithmetics,
Formalized Mathematics 11(4), pages 417-419, 2003. MML Identifier: BINARI_5

**Summary**: In this paper, we have binary arithmetic and its related operations. We include some theorems concerning logical operators.

- Artur Kornilowicz, Yasunari Shidama.
SCMPDS Is Not Standard,
Formalized Mathematics 11(4), pages 421-424, 2003. MML Identifier: SCMPDS_9

**Summary**: The aim of the paper is to show that SCMPDS (\cite{SCMPDS_2.ABS}) does not belong to the class of standard computers (\cite{AMISTD_1.ABS}).

- Robert Milewski.
On the Upper and Lower Approximations of the Curve,
Formalized Mathematics 11(4), pages 425-430, 2003. MML Identifier: JORDAN19

**Summary**:

- Andrzej Trybulec.
On the Sets Inhabited by Numbers,
Formalized Mathematics 11(4), pages 341-347, 2003. MML Identifier: MEMBERED
**Volume 12, 2004**- Number 1
- Yatsuka Nakamura.
Sorting Operators for Finite Sequences,
Formalized Mathematics 12(1), pages 1-4, 2004. MML Identifier: RFINSEQ2

**Summary**: Two kinds of sorting operators, descendent one and ascendent one are introduced for finite sequences of reals. They are also called rearrangement of finite sequences of reals. Maximum and minimum values of finite sequences of reals are also defined. We also discuss relations between these concepts.

- Masaaki Niimura, Yasushi Fuwa.
Magnitude Relation Properties of Radix-$2^k$ SD Number,
Formalized Mathematics 12(1), pages 5-8, 2004. MML Identifier: RADIX_5

**Summary**: In this article, magnitude relation properties of Radix-$2^k$ SD number are discussed. Until now, the Radix-$2^k$ SD Number has been proposed for the high-speed calculations for RSA Cryptograms. In RSA Cryptograms, many modulo calculations are used, and modulo calculations need a comparison between two numbers.\par In this article, we discuss magnitude relation of Radix-$2^k$ SD Number. In the first section, we present some useful theorems for operations of Radix-$2^k$ SD Number. In the second section, we prove some properties of the primary numbers expressed by Radix-$2^k$ SD Number such as 0, 1, and Radix(k). In the third section, we prove primary magnitude relations between two Radix-$2^k$ SD Numbers. In the fourth section, we define Max/Min numbers in some cases. And in the last section, we prove some relations between the addition of Max/Min numbers.

- Masaaki Niimura, Yasushi Fuwa.
High Speed Modulo Calculation Algorithm with Radix-$2^k$ SD Number,
Formalized Mathematics 12(1), pages 9-13, 2004. MML Identifier: RADIX_6

**Summary**: In RSA Cryptograms, many modulo calculations are used, but modulo calculation is based on many subtractions and it takes long a time to calculate it. In this article, we explain a new modulo calculation algorithm using a table. And we prove that upper 3 digits of Radix-$2^k$ SD numbers are enough to specify the answer. \par In the first section, we present some useful theorems for operations of Radix-$2^k$ SD Number. In the second section, we define Upper 3 Digits of Radix-$2^k$ SD number and prove that property. In the third section, we prove some property connected with the minimum digits of Radix-$2^k$ SD number. In the fourth section, we identify the range of modulo arithmetic result and prove that the Upper 3 Digits indicate two possible answers. And in the last section, we define a function to select true answer from the results of Upper 3 Digits.

- Takashi Mitsuishi, Grzegorz Bancerek.
Transitive Closure of Fuzzy Relations,
Formalized Mathematics 12(1), pages 15-20, 2004. MML Identifier: LFUZZY_1

**Summary**:

- Adam Grabowski.
Basic Properties of Rough Sets and Rough Membership Function,
Formalized Mathematics 12(1), pages 21-28, 2004. MML Identifier: ROUGHS_1

**Summary**: We present basic concepts concerning rough set theory. We define tolerance and approximation spaces and rough membership function. Different rough inclusions as well as the predicate of rough equality of sets are also introduced.

- Yatsuka Nakamura.
Correctness of Non Overwriting Programs. Part I,
Formalized Mathematics 12(1), pages 29-32, 2004. MML Identifier: PRGCOR_1

**Summary**: Non overwriting program is a program where each variable used in it is written only just one time, but the control variables used for ``for-statement'' are exceptional. Contrarily, variables are allowed to be read many times. There are other restrictions for the non overwriting program. For statements, only the following are allowed: ``substituting-statement'', ``if-else-statement'', ``for-statement'' (with break and without break), function (correct one) -- ``call-statement'' and ``return-statement''. Grammar of non overwriting program is like the one of the C-language. For type of variables, ``int'', ``real", ``char'' and ``float'' can be used, and array of them can also be used. For operation, ``+'', ``$-$'' and ``*'' are used for a type ``int''; ``+'', ``$-$'', ``*'' and ``/'' are used for a type ``float''. User can also define structures like in C. Non overwriting program can be translated to (predicative) logic formula in definition part to define functions. If a new function is correctly defined, a corresponding program is correct, if it does not use arrays. If it uses arrays, area check is necessary in the following theorem.\par Semantic correctness is shown by some theorems following the definition. These theorems must tie up the result of the program and mathematical concepts introduced before. Correctness is proven {\it function-wise}. We must use only {\it correctness-proven} functions to define a new function (to write a new program as a form of a function). Here, we present two programs of division function of two natural numbers and of two integers. An algorithm is checked for each case by proving correctness of the definitions. We also perform an area check of the index of arrays used in one of the programs.

- Artur Kornilowicz.
A Tree of Execution of a Macroinstruction,
Formalized Mathematics 12(1), pages 33-37, 2004. MML Identifier: AMISTD_3

**Summary**: A tree of execution of a macroinstruction is defined. It is a tree decorated by the instruction locations of a computer. Successors of each vertex are determined by the set of all possible values of the instruction counter after execution of the instruction placed in the location indicated by given vertex.

- Yasunari Shidama.
Banach Space of Bounded Linear Operators,
Formalized Mathematics 12(1), pages 39-48, 2004. MML Identifier: LOPBAN_1

**Summary**: In this article, the basic properties of linear spaces which are defined as the set of all linear operators from one linear space to another, are described. Especially, the Banach space is introduced. This is defined by the set of all bounded linear operators.

- Piotr Rudnicki.
Little Bezout Theorem (Factor Theorem),
Formalized Mathematics 12(1), pages 49-58, 2004. MML Identifier: UPROOTS

**Summary**: We present a formalization of the factor theorem for univariate polynomials, also called the (little) Bezout theorem: Let $r$ belong to a commutative ring $L$ and $p(x)$ be a polynomial over $L$. Then $x-r$ divides $p(x)$ iff $p(r) = 0$. We also prove some consequences of this theorem like that any non zero polynomial of degree $n$ over an algebraically closed integral domain has $n$ (non necessarily distinct) roots.

- Broderick Arneson, Piotr Rudnicki.
Primitive Roots of Unity and Cyclotomic Polynomials,
Formalized Mathematics 12(1), pages 59-67, 2004. MML Identifier: UNIROOTS

**Summary**: We present a formalization of roots of unity, define cyclotomic polynomials and demonstrate the relationship between cyclotomic polynomials and unital polynomials.

- Broderick Arneson, Matthias Baaz, Piotr Rudnicki.
Witt's Proof of the Wedderburn Theorem,
Formalized Mathematics 12(1), pages 69-75, 2004. MML Identifier: WEDDWITT

**Summary**: We present a formalization of Witt's proof of the Wedderburn theorem following Chapter 5 of {\em Proofs from THE BOOK} by Martin Aigner and G\"{u}nter M. Ziegler, 2nd ed., Springer 1999.

- Yatsuka Nakamura.
Sorting Operators for Finite Sequences,
Formalized Mathematics 12(1), pages 1-4, 2004. MML Identifier: RFINSEQ2
- Number 2
- Yasumasa Suzuki.
Banach Space of Bounded Real Sequences,
Formalized Mathematics 12(2), pages 77-83, 2004. MML Identifier: RSSPACE4

**Summary**: We introduce the arithmetic addition and multiplication in the set of bounded real sequences and also introduce the norm. This set has the structure of the Banach space.

- Yuzhong Ding, Xiquan Liang.
Solving Roots of Polynomial Equation of Degree 2 and 3 with Complex Coefficients,
Formalized Mathematics 12(2), pages 85-92, 2004. MML Identifier: POLYEQ_3

**Summary**: In the article, solving complex roots of polynomial equation of degree 2 and 3 with real coefficients and complex roots of polynomial equation of degree 2 and 3 with complex coefficients is discussed.

- Noboru Endou.
Complex Linear Space and Complex Normed Space,
Formalized Mathematics 12(2), pages 93-102, 2004. MML Identifier: CLVECT_1

**Summary**: In this article, we introduce the notion of complex linear space and complex normed space.

- Yasunari Shidama.
The Banach Algebra of Bounded Linear Operators,
Formalized Mathematics 12(2), pages 103-108, 2004. MML Identifier: LOPBAN_2

**Summary**: In this article, the basic properties of Banach algebra are described. This algebra is defined as the set of all bounded linear operators from one normed space to another.

- Noboru Endou.
Complex Linear Space of Complex Sequences,
Formalized Mathematics 12(2), pages 109-117, 2004. MML Identifier: CSSPACE

**Summary**: In this article, we introduce a notion of complex linear space of complex sequence and complex unitary space.

- Yatsuka Nakamura.
Behaviour of an Arc Crossing a Line,
Formalized Mathematics 12(2), pages 119-124, 2004. MML Identifier: JORDAN20

**Summary**: In 2-dimensional Euclidean space, we examine behaviour of an arc when it crosses a vertical line. There are 3 types when an arc enters into a line, which are: ``Left-In'', ``Right-In'' and ``Oscilating-In''. Also, there are 3 types when an arc goes out from a line, which are: ``Left-Out'', ``Right-Out'' and ``Oscilating-Out''. If an arc is a special polygonal arc, there are only 2 types for each case, entering in and going out. They are ``Left-In'' and ``Right-In'' for entering in, and ``Left-Out'' and ``Right-Out'' for going out.

- Masami Tanaka, Yatsuka Nakamura.
Some Set Series in Finite Topological Spaces. Fundamental Concepts for Image Processing,
Formalized Mathematics 12(2), pages 125-129, 2004. MML Identifier: FINTOPO3

**Summary**: First we give a definition of ``inflation'' of a set in finite topological spaces. Then a concept of ``deflation'' of a set is also defined. In the remaining part, we give a concept of the ``set series'' for a subset of a finite topological space. Using this, we can define a series of neighbourhoods for each point in the space. The work is done according to \cite{Nakamura:2}.

- Yasunari Shidama.
The Series on Banach Algebra,
Formalized Mathematics 12(2), pages 131-138, 2004. MML Identifier: LOPBAN_3

**Summary**: In this article, the basic properties of the series on Banach algebra are described. The Neumann series is introduced in the last section.

- Pacharapokin Chanapat, Kanchun , Hiroshi Yamazaki.
Formulas and Identities of Trigonometric Functions,
Formalized Mathematics 12(2), pages 139-141, 2004. MML Identifier: SIN_COS4

**Summary**: In this article, we concentrated especially on addition formulas of fundamental trigonometric functions, and their identities.

- Krzysztof Retel.
The Class of Series-Parallel Graphs. Part III,
Formalized Mathematics 12(2), pages 143-149, 2004. MML Identifier: NECKLA_3

**Summary**: This paper contains some facts and theorems relating to the following operations on graphs: union, sum, complement and ``embeds''. We also introduce connected graphs to prove that a finite irreflexive symmetric N-free graph is a finite series-parallel graph. This article continues the formalization of \cite{Thomasse}.

- Artur Kornilowicz, Yasunari Shidama.
Relocability for SCM over Ring,
Formalized Mathematics 12(2), pages 151-157, 2004. MML Identifier: SCMRING4

**Summary**:

- Noboru Endou.
Convergent Sequences in Complex Unitary Space,
Formalized Mathematics 12(2), pages 159-165, 2004. MML Identifier: CLVECT_2

**Summary**: In this article, we introduce the notion of convergence sequence in complex unitary space and complex Hilbert space.

- Artur Kornilowicz.
Recursive Definitions. Part II,
Formalized Mathematics 12(2), pages 167-172, 2004. MML Identifier: RECDEF_2

**Summary**:

- Yasunari Shidama.
The Exponential Function on Banach Algebra,
Formalized Mathematics 12(2), pages 173-177, 2004. MML Identifier: LOPBAN_4

**Summary**: In this article, the basic properties of the exponential function on Banach algebra are described.

- Artur Kornilowicz, Piotr Rudnicki.
Fundamental Theorem of Arithmetic,
Formalized Mathematics 12(2), pages 179-186, 2004. MML Identifier: NAT_3

**Summary**: We formalize the notion of the prime-power factorization of a natural number and prove the Fundamental Theorem of Arithmetic. We prove also how prime-power factorization can be used to compute: products, quotients, powers, greatest common divisors and least common multiples.

- Noboru Endou.
Hilbert Space of Complex Sequences,
Formalized Mathematics 12(2), pages 187-190, 2004. MML Identifier: CSSPACE2

**Summary**: An extension of \cite{RSSPACE2.ABS}. As the example of complex norm spaces, we introduce the arithmetic addition and multiplication in the set of absolute summable complex sequences and also introduce the norm.

- Noboru Endou.
Banach Space of Absolute Summable Complex Sequences,
Formalized Mathematics 12(2), pages 191-194, 2004. MML Identifier: CSSPACE3

**Summary**: An extension of \cite{RSSPACE3.ABS}. As the example of complex norm spaces, I introduced the arithmetic addition and multiplication in the set of absolute summable complex sequences and also introduced the norm.

- Yasunari Shidama.
The Taylor Expansions,
Formalized Mathematics 12(2), pages 195-200, 2004. MML Identifier: TAYLOR_1

**Summary**: In this article, some classic theorems of calculus are described. The Taylor expansions and the logarithmic differentiation, etc. are included here.

- Noboru Endou.
Complex Banach Space of Bounded Linear Operators,
Formalized Mathematics 12(2), pages 201-209, 2004. MML Identifier: CLOPBAN1

**Summary**: An extension of \cite{LOPBAN_1.ABS}. In this article, the basic properties of complex linear spaces which are defined by the set of all complex linear operators from one complex linear space to another are described. Finally, a complex Banach space is introduced. This is defined by the set of all bounded complex linear operators, like in \cite{LOPBAN_1.ABS}.

- Noboru Endou.
Complex Banach Space of Bounded Complex Sequences,
Formalized Mathematics 12(2), pages 211-218, 2004. MML Identifier: CSSPACE4

**Summary**: An extension of \cite{RSSPACE4.ABS}. In this article, we introduce two complex Banach spaces. One of them is the space of bounded complex sequences. The other one is the space of complex bounded functions, which is defined by the set of all complex bounded functions.

- Hirofumi Fukura, Yatsuka Nakamura.
Concatenation of Finite Sequences Reducing Overlapping Part and an Argument of Separators of Sequential Files,
Formalized Mathematics 12(2), pages 219-224, 2004. MML Identifier: FINSEQ_8

**Summary**: For two finite sequences, we present a notion of their concatenation, reducing overlapping part of the tail of the former and the head of the latter. At the same time, we also give a notion of common part of two finite sequences, which relates to the concatenation given here. A finite sequence is separated by another finite sequence (separator). We examined the condition that a separator separates uniquely any finite sequence. This will become a model of a separator of sequential files.

- Yasumasa Suzuki, Noboru Endou.
Cauchy Sequence of Complex Unitary Space,
Formalized Mathematics 12(2), pages 225-229, 2004. MML Identifier: CLVECT_3

**Summary**: As an extension of \cite{BHSP_4.ABS}, we introduce the Cauchy sequence of complex unitary space and describe its properties.

- Yasumasa Suzuki.
Banach Space of Bounded Real Sequences,
Formalized Mathematics 12(2), pages 77-83, 2004. MML Identifier: RSSPACE4
- Number 3
- Noboru Endou.
Complex Valued Functions Space,
Formalized Mathematics 12(3), pages 231-235, 2004. MML Identifier: CFUNCDOM

**Summary**: This article is an extension of \cite{FUNCSDOM.ABS} to complex valued functions.

- Noboru Endou.
Banach Algebra of Bounded Complex Linear Operators,
Formalized Mathematics 12(3), pages 237-242, 2004. MML Identifier: CLOPBAN2

**Summary**: This article is an extension of \cite{LOPBAN_2.ABS}.

- Yuzhong Ding, Xiquan Liang.
Formulas and Identities of Trigonometric Functions,
Formalized Mathematics 12(3), pages 243-246, 2004. MML Identifier: SIN_COS5

**Summary**:

- Yuzhong Ding, Xiquan Liang.
Solving Roots of the Special Polynomial Equation with Real Coefficients,
Formalized Mathematics 12(3), pages 247-250, 2004. MML Identifier: POLYEQ_4

**Summary**:

- Adam Grabowski, Artur Kornilowicz.
Algebraic Properties of Homotopies,
Formalized Mathematics 12(3), pages 251-260, 2004. MML Identifier: BORSUK_6

**Summary**:

- Artur Kornilowicz, Yasunari Shidama, Adam Grabowski.
The Fundamental Group,
Formalized Mathematics 12(3), pages 261-268, 2004. MML Identifier: TOPALG_1

**Summary**: This is the next article in a series devoted to homotopy theory (following \cite{BORSUK_2.ABS} and \cite{BORSUK_6.ABS}). The concept of fundamental groups of pointed topological spaces has been introduced. Isomorphism of fundamental groups defined with respect to different points belonging to the same component has been stated. Triviality of fundamental group(s) of ${\Bbb R}^n$ has been shown.

- Takaya Nishiyama, Keiji Ohkubo, Yasunari Shidama.
The Continuous Functions on Normed Linear Spaces,
Formalized Mathematics 12(3), pages 269-275, 2004. MML Identifier: NFCONT_1

**Summary**: In this article, the basic properties of the continuous function on normed linear spaces are described.

- Takaya Nishiyama, Artur Kornilowicz, Yasunari Shidama.
The Uniform Continuity of Functions on Normed Linear Spaces,
Formalized Mathematics 12(3), pages 277-279, 2004. MML Identifier: NFCONT_2

**Summary**: In this article, the basic properties of uniform continuity of functions on normed linear spaces are described.

- Noboru Endou.
Series on Complex Banach Algebra,
Formalized Mathematics 12(3), pages 281-288, 2004. MML Identifier: CLOPBAN3

**Summary**: This article is an extension of \cite{LOPBAN_3.ABS}.

- Noboru Endou.
Exponential Function on Complex Banach Algebra,
Formalized Mathematics 12(3), pages 289-293, 2004. MML Identifier: CLOPBAN4

**Summary**: This article is an extension of \cite{LOPBAN_4.ABS}.

- Artur Kornilowicz.
The Fundamental Group of Convex Subspaces of $\calE^n_\rmT$,
Formalized Mathematics 12(3), pages 295-299, 2004. MML Identifier: TOPALG_2

**Summary**: The triviality of the fundamental group of subspaces of ${\cal E}^n_{\rm T}$ and ${\Bbb R}^{\bf 1}$ have been shown.

- Artur Kornilowicz, Yasunari Shidama.
Intersections of Intervals and Balls in $\calE^n_\rmT$,
Formalized Mathematics 12(3), pages 301-306, 2004. MML Identifier: TOPREAL9

**Summary**:

- Magdalena Jastrzcebska, Adam Grabowski.
Some Properties of Fibonacci Numbers,
Formalized Mathematics 12(3), pages 307-313, 2004. MML Identifier: FIB_NUM2

**Summary**: We formalized some basic properties of the Fibonacci numbers using definitions and lemmas from \cite{PRE_FF.ABS} and \cite{FIB_NUM.ABS}, e.g. Cassini's and Catalan's identities. We also showed the connections between Fibonacci numbers and Pythagorean triples as defined in \cite{PYTHTRIP.ABS}. The main result of this article is a proof of Carmichael Theorem on prime divisors of prime-generated Fibonacci numbers. According to it if we look at the prime factors of a Fibonacci number generated by a prime number, none of them has appeared as a factor in any earlier Fibonacci number. We plan to develop the full proof of the Carmichael Theorem following \cite{Yabuta:01}.

- Ewa Romanowicz, Adam Grabowski.
The Hall Marriage Theorem,
Formalized Mathematics 12(3), pages 315-320, 2004. MML Identifier: HALLMAR1

**Summary**: The Marriage Theorem, as credited to Philip Hall \cite{Hall:1935}, gives the necessary and sufficient condition allowing us to select a distinct element from each of a finite collection $\{A_i\}$ of $n$ finite subsets. This selection, called a set of different representatives (SDR), exists if and only if the marriage condition (or Hall condition) is satisfied: $$\forall_{J\subseteq\{1,\dots,n\}}|\bigcup_{i\in J} A_i|\geq |J|.$$ The proof which is given in this article (according to Richard Rado, 1967) is based on the lemma that for finite sequences with non-trivial elements which satisfy Hall property there exists a reduction (see Def. 5) such that Hall property again holds (see Th.~29 for details).

- Hiroshi Imura, Morishige Kimura, Yasunari Shidama.
The Differentiable Functions on Normed Linear Spaces,
Formalized Mathematics 12(3), pages 321-327, 2004. MML Identifier: NDIFF_1

**Summary**: In this article, the basic properties of the differentiable functions on normed linear spaces are described.

- Piotr Wojtecki, Adam Grabowski.
Lucas Numbers and Generalized Fibonacci Numbers,
Formalized Mathematics 12(3), pages 329-333, 2004. MML Identifier: FIB_NUM3

**Summary**: The recursive definition of Fibonacci sequences \cite{PRE_FF.ABS} is a good starting point for various variants and generalizations. We can point out here e.g. Lucas (with $2$ and $1$ as opening values) and the so-called generalized Fibonacci numbers (starting with arbitrary integers $a$ and $b$). \par In this paper, we introduce Lucas and G-numbers and we state their basic properties analogous to those proven in \cite{FIB_NUM.ABS} and \cite{FIB_NUM2.ABS}.

- Katarzyna Romanowicz, Adam Grabowski.
The Operation of Addition of Relational Structures,
Formalized Mathematics 12(3), pages 335-339, 2004. MML Identifier: LATSUM_1

**Summary**: The article contains the formalization of the addition operator on relational structures as defined by A.~Wro{\'n}ski \cite{Wronski:1974} (as a generalization of Troelstra's sum or Ja{\'s}kowski star addition). The ordering relation of $A \otimes B$ is given by $$\le_{A\otimes B}\:=\:\le_A\cup \le_B\cup\: (\le_A \circ \le_B),$$ where the carrier is defined as the set-theoretical union of carriers of $A$ and $B$. Main part -- Section 3 -- is devoted to the Mizar translation of Theorem 1 (iv--xiii), p.~66 of \cite{Wronski:1974}.

- Karol Pcak.
The Nagata-Smirnov Theorem. Part I,
Formalized Mathematics 12(3), pages 341-346, 2004. MML Identifier: NAGATA_1

**Summary**: In this paper we define a discrete subset family of a topological space and basis sigma locally finite and sigma discrete. We prove first an auxiliary fact for discrete family and sigma locally finite and sigma discrete basis. We show also the necessary condition for the Nagata Smirnov theorem: every metrizable space is $T_3$ and has a sigma locally finite basis. Also, we define a sufficient condition for a $T_3$ topological space to be $T_4$. We introduce the concept of pseudo metric.

- Gijs Geleijnse, Grzegorz Bancerek.
Properties of Groups,
Formalized Mathematics 12(3), pages 347-350, 2004. MML Identifier: GROUP_8

**Summary**: In this article we formalize theorems from Chapter 1 of \cite{Hall:1959}. Our article covers Theorems 1.5.4, 1.5.5 (inequality on indices), 1.5.6 (equality of indices), Lemma 1.6.1 and several other supporting theorems needed to complete the formalization.

- Dorota Czcestochowska, Adam Grabowski.
Catalan Numbers,
Formalized Mathematics 12(3), pages 351-353, 2004. MML Identifier: CATALAN1

**Summary**: In this paper, we define Catalan sequence (starting from $0$) and prove some of its basic properties. The Catalan numbers ($0,1,1,2,5,14,42,\dots$) arise in a number of problems in combinatorics. They can be computed e.g. using the formula $$C_n=\frac{{{2n}\choose {n}}}{n+1},$$ their recursive definition is also well known: $$C_1=1,\quad C_n=\Sigma_{i=1}^{n-1}C_i C_{n-i},\quad n\geq 2.$$ Among other things, the Catalan numbers describe the number of ways in which parentheses can be placed in a sequence of numbers to be multiplied, two at a time.

- Violetta Kozarkiewicz, Adam Grabowski.
Axiomatization of Boolean Algebras Based on Sheffer Stroke,
Formalized Mathematics 12(3), pages 355-361, 2004. MML Identifier: SHEFFER1

**Summary**: We formalized another axiomatization of Boolean algebras. The classical one, as introduced in \cite{LATTICES.ABS}, ``the fourth set of postulates'' due to Huntington \cite{Huntington1} (\cite{ROBBINS1.ABS} in Mizar) and the single axiom in terms of disjunction and negation \cite{ROBBINS2.ABS}. In this article, we aimed at the description of Boolean algebras using Sheffer stroke according to \cite{Sheffer:1913}, namely by the following three axioms: $$(x

- Aneta Lukaszuk, Adam Grabowski.
Short Sheffer Stroke-Based Single Axiom for Boolean Algebras,
Formalized Mathematics 12(3), pages 363-370, 2004. MML Identifier: SHEFFER2

**Summary**: We continue the description of Boolean algebras in terms of the Sheffer stroke as defined in \cite{SHEFFER1.ABS}. The single axiomatization for BAs in terms of disjunction and negation was shown in \cite{ROBBINS2.ABS}. As was checked automatically with the help of automated theorem prover Otter, single axiom of the form $$(x

- Hiroshi Imura, Yuji Sakai, Yasunari Shidama.
Differentiable Functions on Normed Linear Spaces. Part II,
Formalized Mathematics 12(3), pages 371-374, 2004. MML Identifier: NDIFF_2

**Summary**: A continuation of \cite{NDIFF_1.ABS}, the basic properties of the differentiable functions on normed linear spaces are described.

- Takaya Nishiyama, Hirofumi Fukura, Yatsuka Nakamura.
Logical Correctness of Vector Calculation Programs,
Formalized Mathematics 12(3), pages 375-380, 2004. MML Identifier: PRGCOR_2

**Summary**: In C-program, vectors of $n$-dimension are sometimes represented by arrays, where the dimension n is saved in the 0-th element of each array. If we write the program in non-overwriting type, we can gi Here, we give a program calculating inner product of 2 vectors, as an example of such a type, and its Logical-Model. If the Logical-Model is well defined, and theorems tying the model with previous definitions are given, we can say that the program is correct logically. In case the program is given as implicit function form (i.e., the result of calculation is given by a variable of one of arguments of a function), its Logical-Model is given by a definition of a ne Logical correctness of such a program is shown by theorems following the definition. As examples of such programs, we presented vector calculation of add, sub, minus and scalar product.

- Hiroshi Imura, Masami Tanaka, Yatsuka Nakamura.
Continuous Mappings between Finite and One-Dimensional Finite Topological Spaces,
Formalized Mathematics 12(3), pages 381-384, 2004. MML Identifier: FINTOPO4

**Summary**: We showed relations between separateness and inflation operation. We also gave some relations between separateness and connectedness defined before. For two finite topological spaces, we defined a continuous function from one to another. Some topological concepts are preserved by such continuous functions. We gave one-dimensional concrete models of finite topological space.

- Karol Pcak.
The Nagata-Smirnov Theorem. Part II,
Formalized Mathematics 12(3), pages 385-389, 2004. MML Identifier: NAGATA_2

**Summary**: In this paper we show some auxiliary facts for sequence function to be pseudo-metric. Next we prove the Nagata-Smirnov theorem that every topological space is metrizable if and only if it has $\sigma$-locally finite basis. We attach also the proof of the Bing's theorem that every topological space is metrizable if and only if its basis is $\sigma$-discrete.

- Artur Kornilowicz.
On the Isomorphism of Fundamental Groups,
Formalized Mathematics 12(3), pages 391-396, 2004. MML Identifier: TOPALG_3

**Summary**:

- Noboru Endou.
Algebra of Complex Vector Valued Functions,
Formalized Mathematics 12(3), pages 397-401, 2004. MML Identifier: VFUNCT_2

**Summary**: This article is an extension of \cite{VFUNCT_1.ABS}.

- Noboru Endou.
Continuous Functions on Real and Complex Normed Linear Spaces,
Formalized Mathematics 12(3), pages 403-419, 2004. MML Identifier: NCFCONT1

**Summary**: This article is an extension of \cite{NFCONT_1.ABS}.

- Artur Kornilowicz.
On the Fundamental Groups of Products of Topological Spaces,
Formalized Mathematics 12(3), pages 421-425, 2004. MML Identifier: TOPALG_4

**Summary**: In the paper we show that fundamental group of the product of two topological spaces is isomorphic to the product of fundamental groups of the spaces.

- Noboru Endou.
Complex Valued Functions Space,
Formalized Mathematics 12(3), pages 231-235, 2004. MML Identifier: CFUNCDOM
**Volume 13, 2005**- Number 1
- Ming Liang, Yuzhong Ding.
Partial Sum of Some Series,
Formalized Mathematics 13(1), pages 1-4, 2005. MML Identifier: SERIES_2

**Summary**: Solving the partial sum of some often used series.

- Patrick Braselmann, Peter Koepke.
Substitution in First-Order Formulas: Elementary Properties,
Formalized Mathematics 13(1), pages 5-15, 2005. MML Identifier: SUBSTUT1

**Summary**: This article is part of a series of Mizar articles which constitute a formal proof (of a basic version) of Kurt G{\"o}del's famous completeness theorem (K. G{\"o}del, ``Die Vollst{\"a}ndigkeit der Axiome des logischen Funktionenkalk{\"u}ls'', Monatshefte f\"ur Mathematik und Physik 37 (1930), 349-360). The completeness theorem provides the theoretical basis for a uniform formalization of mathematics as in the Mizar project. We formalize first-order logic up to the completeness theorem as in H. D. Ebbinghaus, J. Flum, and W. Thomas, Mathematical Logic, 1984, Springer Verlag New York Inc. The present article introduces the basic concepts of substitution of a variable for a variable in a first-order formula. The contents of this article correspond to Chapter III par. 8, Definition 8.1, 8.2 of Ebbinghaus, Flum, Thomas.

- Patrick Braselmann, Peter Koepke.
Coincidence Lemma and Substitution Lemma,
Formalized Mathematics 13(1), pages 17-26, 2005. MML Identifier: SUBLEMMA

**Summary**: This article is part of a series of Mizar articles which constitute a formal proof (of a basic version) of Kurt G{\"o}del's famous completeness theorem (K. G{\"o}del, ``Die Vollst{\"a}ndigkeit der Axiome des logischen Funktionenkalk{\"u}ls'', Monatshefte f\"ur Mathematik und Physik 37 (1930), 349--360). The completeness theorem provides the theoretical basis for a uniform formalization of mathematics as in the Mizar project. We formalize first-order logic up to the completeness theorem as in H. D. Ebbinghaus, J. Flum, and W. Thomas, Mathematical Logic, 1984, Springer Verlag New York Inc. The present article establishes further concepts of substitution of a variable for a variable in a first-order formula. The main result is the substitution lemma. The contents of this article correspond to Chapter III par. 5, 5.1 Coincidence Lemma and Chapter III par. 8, 8.3 Substitution Lemma of Ebbinghaus, Flum, Thomas.

- Patrick Braselmann, Peter Koepke.
Substitution in First-Order Formulas. Part II. The Construction of First-Order Formulas,
Formalized Mathematics 13(1), pages 27-32, 2005. MML Identifier: SUBSTUT2

**Summary**: This article is part of a series of Mizar articles which constitute a formal proof (of a basic version) of Kurt G{\"o}del's famous completeness theorem (K. G{\"o}del, ``Die Vollst{\"a}ndigkeit der Axiome des logischen Funktionenkalk{\"u}ls'', Monatshefte f\"ur Mathematik und Physik 37 (1930), 349-360). The completeness theorem provides the theoretical basis for a uniform formalization of mathematics as in the Mizar project. We formalize first-order logic up to the completeness theorem as in H. D. Ebbinghaus, J. Flum, and W. Thomas, Mathematical Logic, 1984, Springer Verlag New York Inc. The present article establishes that every substitution can be applied to every formula as in Chapter III par. 8, Definition 8.1, 8.2 of Ebbinghaus, Flum, Thomas. After that, it is observed that substitution doesn't change the number of quantifiers of a formula. Then further details about substitution and some results about the construction of formulas are proven.

- Patrick Braselmann, Peter Koepke.
A Sequent Calculus for First-Order Logic,
Formalized Mathematics 13(1), pages 33-39, 2005. MML Identifier: CALCUL_1

**Summary**: This article is part of a series of Mizar articles which constitute a formal proof (of a basic version) of Kurt G{\"o}del's famous completeness theorem (K. G{\"o}del, ``Die Vollst{\"a}ndigkeit der Axiome des logischen Funktionenkalk{\"u}ls'', Monatshefte f\"ur Mathematik und Physik 37 (1930), 349--360). The completeness theorem provides the theoretical basis for a uniform formalization of mathematics as in the Mizar project. We formalize first-order logic up to the completeness theorem as in H. D. Ebbinghaus, J. Flum, and W. Thomas, Mathematical Logic, 1984, Springer Verlag New York Inc. The present article introduces a sequent calculus for first-order logic. The correctness of this calculus is shown and some important inferences are derived. The contents of this article correspond to Chapter IV of Ebbinghaus, Flum, Thomas.

- Patrick Braselmann, Peter Koepke.
Consequences of the Sequent Calculus,
Formalized Mathematics 13(1), pages 41-44, 2005. MML Identifier: CALCUL_2

**Summary**: This article is part of a series of Mizar articles which constitute a formal proof (of a basic version) of Kurt G{\"{o}}del's famous completeness theorem (K. G{\"{o}}del, ``Die Vollst{\"{a}}ndigkeit der Axiome des logischen Funktionenkalk{\"{u}}ls'', Monatshefte f\"ur Mathematik und Physik 37 (1930), 349-360). The completeness theorem provides the theoretical basis for a uniform formalization of mathematics as in the Mizar project. We formalize first-order logic up to the completeness theorem as in H. D. Ebbinghaus, J. Flum, and W. Thomas, Mathematical Logic, 1984, Springer Verlag New York Inc. The first main result of the present article is that the derivablility of a sequent doesn't depend on the ordering of the antecedent. The second main result says: if a sequent is derivable, then the formulas in the antecendent only need to occur once.

- Patrick Braselmann, Peter Koepke.
Equivalences of Inconsistency and Henkin Models,
Formalized Mathematics 13(1), pages 45-48, 2005. MML Identifier: HENMODEL

**Summary**: This article is part of a series of Mizar articles which constitute a formal proof (of a basic version) of Kurt G{\"{o}}del's famous completeness theorem (K. G{\"{o}}del, ``Die Vollst{\"{a}}ndigkeit der Axiome des logischen Funktionenkalk{\"{u}}ls'', Monatshefte f\"ur Mathematik und Physik 37 (1930), 349--360). The completeness theorem provides the theoretical basis for a uniform formalization of mathematics as in the Mizar project. We formalize first-order logic up to the completeness theorem as in H. D. Ebbinghaus, J. Flum, and W. Thomas, Mathematical Logic, 1984, Springer Verlag, New York Inc. The present article establishes some equivalences of inconsistency. It is proved that a countable union of consistent sets is consistent. Then the concept of a Henkin model is introduced. The contents of this article correspond to Chapter IV, par. 7 and Chapter V, par. 1 of Ebbinghaus, Flum, Thomas.

- Patrick Braselmann, Peter Koepke.
G\"odel's Completeness Theorem,
Formalized Mathematics 13(1), pages 49-53, 2005. MML Identifier: GOEDELCP

**Summary**: This article is part of a series of Mizar articles which constitute a formal proof (of a basic version) of Kurt G{\"o}del's famous completeness theorem (K. G{\"o}del, ``Die Vollst{\"a}ndigkeit der Axiome des logischen Funktionenkalk{\"u}ls'', Monatshefte f\"ur Mathematik und Physik 37 (1930), 349--360). The completeness theorem provides the theoretical basis for a uniform formalization of mathematics as in the Mizar project. We formalize first-order logic up to the completeness theorem as in H. D. Ebbinghaus, J. Flum, and W. Thomas, Mathematical Logic, 1984, Springer Verlag New York Inc. The present article contains the proof of a simplified completeness theorem for a countable relational language without equality.

- Shunichi Kobayashi.
Propositional Calculus for Boolean Valued Functions. Part VIII,
Formalized Mathematics 13(1), pages 55-58, 2005. MML Identifier: BVFUNC26

**Summary**: In this paper, we proved some elementary propositional calculus formulae for Boolean valued functions.

- Yasumasa Suzuki.
H\"older's Inequality and Minkowski's Inequality,
Formalized Mathematics 13(1), pages 59-61, 2005. MML Identifier: HOLDER_1

**Summary**: In this article, H\"older's inequality and Minkowski's inequality are proved. These equalities are basic ones of functional analysis.

- Yasumasa Suzuki.
The Banach Space $l^p$,
Formalized Mathematics 13(1), pages 63-66, 2005. MML Identifier: LP_SPACE

**Summary**: We introduce the arithmetic addition and multiplication in the set of $l^p$ real sequences and also introduce the norm. This set has the structure of the Banach space.

- Yasunari Shidama, Noboru Endou.
Lebesgue Integral of Simple Valued Function,
Formalized Mathematics 13(1), pages 67-71, 2005. MML Identifier: MESFUNC3

**Summary**: In this article, the authors introduce Lebesgue integral of simple valued function.

- Artur Kornilowicz, Yasunari Shidama.
Inverse Trigonometric Functions Arcsin and Arccos,
Formalized Mathematics 13(1), pages 73-79, 2005. MML Identifier: SIN_COS6

**Summary**: Notions of inverse sine and inverse cosine have been introduced. Their basic properties have been proved.

- Artur Kornilowicz.
On Some Points of a Simple Closed Curve,
Formalized Mathematics 13(1), pages 81-87, 2005. MML Identifier: JORDAN21

**Summary**:

- Artur Kornilowicz, Adam Grabowski.
On Some Points of a Simple Closed Curve. Part II,
Formalized Mathematics 13(1), pages 89-91, 2005. MML Identifier: JORDAN22

**Summary**: In the paper we formalize some lemmas needed by the proof of the Jordan Curve Theorem according to \cite{TAKE-NAKA}. We show basic properties of the upper and the lower approximations of a simple closed curve (as its compactness and connectedness) and some facts about special points of such approximations.

- Noboru Endou.
Uniform Continuity of Functions on Normed Complex Linear Spaces,
Formalized Mathematics 13(1), pages 93-98, 2005. MML Identifier: NCFCONT2

**Summary**:

- Czeslaw Bylinski.
Introduction to Real Linear Topological Spaces,
Formalized Mathematics 13(1), pages 99-107, 2005. MML Identifier: RLTOPSP1

**Summary**:

- Artur Kornilowicz, Yasunari Shidama.
Some Properties of Rectangles on the Plane,
Formalized Mathematics 13(1), pages 109-115, 2005. MML Identifier: TOPREALA

**Summary**:

- Artur Kornilowicz, Yasunari Shidama.
Some Properties of Circles on the Plane,
Formalized Mathematics 13(1), pages 117-124, 2005. MML Identifier: TOPREALB

**Summary**:

- Adam Naumowicz.
On the Characterization of Collineations of the Segre Product of Strongly Connected Partial Linear Spaces,
Formalized Mathematics 13(1), pages 125-131, 2005. MML Identifier: PENCIL_3

**Summary**: In this paper we characterize the automorphisms (collineations) of the Segre product of partial linear spaces. In particular, we show that if all components of the product are strongly connected, then every collineation is determined by a set of isomorphisms between its components. The formalization follows the ideas presented in the {\em Journal of Geometry} paper \cite{ANKPJG} by Naumowicz and Pra\.zmowski.

- Adam Naumowicz.
Spaces of Pencils, Grassmann Spaces, and Generalized Veronese Spaces,
Formalized Mathematics 13(1), pages 133-138, 2005. MML Identifier: PENCIL_4

**Summary**: In this paper we construct several examples of partial linear spaces. First, we define two algebraic structures, namely the spaces of $k$-pencils and Grassmann spaces for vector spaces over an arbitrary field. Then we introduce the notion of generalized Veronese spaces following the definition presented in the paper \cite{ANKPRM} by Naumowicz and Pra\.zmowski. For all spaces defined, we state the conditions under which they are not degenerated to a single line.

- Adam Grabowski.
On the Boundary and Derivative of a Set,
Formalized Mathematics 13(1), pages 139-146, 2005. MML Identifier: TOPGEN_1

**Summary**: This is the first Mizar article in a series aiming at a complete formalization of the textbook ``General Topology'' by Engelking \cite{ENGEL:1}. We cover the first part of Section 1.3, by defining such notions as a derivative of a subset $A$ of a topological space (usually denoted by $A^{\rm d}$, but ${\rm Der~} A$ in our notation), the derivative and the boundary of families of subsets, points of accumulation and isolated points. We also introduce dense-in-itself, perfect and scattered topological spaces and formulate the notion of the density of a space. Some basic properties are given as well as selected exercises from \cite{ENGEL:1}.

- Christoph Schwarzweller.
Construction of Gr\"obner Bases: Avoiding S-Polynomials -- Buchberger's First Criterium,
Formalized Mathematics 13(1), pages 147-156, 2005. MML Identifier: GROEB_3

**Summary**: We continue the formalization of Groebner bases following the book ``Groebner Bases -- A Computational Approach to Commutative Algebra'' by Becker and Weispfenning. Here we prove Buchberger's first criterium on avoiding S-polynomials: S-polynomials for polynomials with disjoint head terms need not be considered when constructing Groebner bases. In the course of formalizing this theorem we also introduced the splitting of a polynomial in an upper and a lower polynomial containing the greater resp. smaller terms of the original polynomial with respect to a given term order.

- Wenpai Chang, Hiroshi Yamazaki, Yatsuka Nakamura.
A Theory of Matrices of Complex Elements,
Formalized Mathematics 13(1), pages 157-162, 2005. MML Identifier: MATRIX_5

**Summary**: A concept of ``Matrix of Complex'' is defined here. Addition, subtraction, scalar multiplication and product are introduced using correspondent definitions of ``Matrix of Field''. Many equations for such operations consist of a case of ``Matrix of Field''. A calculation method of product of matrices is shown using a finite sequence of Complex in the last theorem.

- Grzegorz Bancerek.
On the Characteristic and Weight of a Topological Space,
Formalized Mathematics 13(1), pages 163-169, 2005. MML Identifier: TOPGEN_2

**Summary**: We continue Mizar formalization of General Topology according to the book \cite{ENGEL:1} by Engelking. In the article the formalization of Section 1.1 is completed. Namely, the paper includes the formalization of theorems on correspondence of the basis and basis in a point, definitions of the character of a point and a topological space, a neighborhood system, and the weight of a topological space. The formalization is tested with almost discrete topological spaces with infinity.

- Grzegorz Bancerek.
On Constructing Topological Spaces and Sorgenfrey Line,
Formalized Mathematics 13(1), pages 171-179, 2005. MML Identifier: TOPGEN_3

**Summary**: We continue Mizar formalization of General Topology according to the book \cite{ENGEL:1} by Engelking. In the article the formalization of Section 1.2 is almost completed. Namely, we formalize theorems on introduction of topologies by bases, neighborhood systems, closed sets, closure operator, and interior operator. The Sorgenfrey line is defined by a basis. It is proved that the weight of it is continuum. Other techniques are used to demonstrate introduction of discrete and anti-discrete topologies.

- Artur Kornilowicz.
On the Real Valued Functions,
Formalized Mathematics 13(1), pages 181-187, 2005. MML Identifier: PARTFUN3

**Summary**:

- Adam Grabowski, Markus Moschner.
Formalization of Ortholattices via~Orthoposets,
Formalized Mathematics 13(1), pages 189-197, 2005. MML Identifier: ROBBINS3

**Summary**: There are two approaches to lattices used in the Mizar Mathematical Library: on the one hand, these structures are based on the set with two binary operations (with an equational characterization as in \cite{LATTICES.ABS}). On the other hand, we may look at them as at relational structures (posets -- see \cite{ORDERS_1.ABS}). As the main result of this article we can state that the Mizar formalization enables us to use both approaches simultaneously (Section 3). This is especially useful because most of lemmas on ortholattices in the literature are stated in the poset setting, so we cannot use equational theorem provers in a straightforward way. We give also short equational characterization of lattices via four axioms (as it was done in \cite{McCune:2005} with the help of the Otter prover). Some corresponding results about ortholattices are also formalized.

- Ming Liang, Yuzhong Ding.
Partial Sum of Some Series,
Formalized Mathematics 13(1), pages 1-4, 2005. MML Identifier: SERIES_2
- Number 2
- Yatsuka Nakamura, Andrzej Trybulec.
The Fashoda Meet Theorem for Rectangles,
Formalized Mathematics 13(2), pages 199-219, 2005. MML Identifier: JGRAPH_7

**Summary**: Here, so called Fashoda Meet Theorem is proven in the case of rectangles. All cases of proper location of arcs are listed up, and it is shown that the theorem consists in each case. Such a list of cases will be useful when one wants to apply the theorem.

- Yuzhong Ding, Xiquan Liang.
Preliminaries to Mathematical Morphology and Its Properties,
Formalized Mathematics 13(2), pages 221-225, 2005. MML Identifier: MATHMORP

**Summary**: In this paper we have discussed the basic mathematical morphological operators and their properties.

- Robert Milewski.
Subsequences of Almost, Weakly and Poorly One-to-one Finite Sequences,
Formalized Mathematics 13(2), pages 227-233, 2005. MML Identifier: JORDAN23

**Summary**: {}

- Gilbert Lee, Piotr Rudnicki.
Alternative Graph Structures,
Formalized Mathematics 13(2), pages 235-252, 2005. MML Identifier: GLIB_000

**Summary**: We define the notion of a graph anew without using the available Mizar structures. In our approach, we model graph structure as a finite function whose domain is a subset of natural numbers. The elements of the domain of the function play the role of selectors for accessing the components of the structure. As these selectors are first class objects, many future extensions of the new graph structure turned out to be easier to formalize in Mizar than with the traditional Mizar structures. \par After introducing graph structure, we define its selectors and then conditions that the structure needs to satisfy to form a directed graph (in the spirit of \cite{GRAPH_1.ABS}). For these graphs we define a collection of basic graph notions; the presentation of these notions is continued in articles \cite{GLIB_001.ABS,GLIB_002.ABS,GLIB_003.ABS}. \par We have tried to follow a number of graph theory books in choosing graph terminology but since the terminology is not commonly agreed upon, we had to make a number of compromises, see \cite{gl04}.

- Gilbert Lee.
Walks in Graphs,
Formalized Mathematics 13(2), pages 253-269, 2005. MML Identifier: GLIB_001

**Summary**: We define walks for graphs introduced in \cite{GLIB_000.ABS}, introduce walk attributes and functors for walk creation and modification of walks. Subwalks of a walk are also defined.

- Gilbert Lee.
Trees and Graph Components,
Formalized Mathematics 13(2), pages 271-277, 2005. MML Identifier: GLIB_002

**Summary**: In the graph framework of \cite{GLIB_000.ABS} we define connected and acyclic graphs, components of a graph, and define the notion of cut-vertex (articulation point).

- Gilbert Lee.
Weighted and Labeled Graphs,
Formalized Mathematics 13(2), pages 279-293, 2005. MML Identifier: GLIB_003

**Summary**: In the graph framework of \cite{GLIB_000.ABS} we introduce new selectors: weights for edges and lables for both edges and vertices. We introduce also a number of tools for accessing and modifying these new fields.

- Gilbert Lee, Piotr Rudnicki.
Correctness of Dijkstra's Shortest Path and Prim's Minimum Spanning Tree Algorithms,
Formalized Mathematics 13(2), pages 295-304, 2005. MML Identifier: GLIB_004

**Summary**: We prove correctness for Dijkstra's shortest path algorithm, and Prim's minimum weight spanning tree algorithm at the level of graph manipulations.

- Gilbert Lee.
Correctnesss of Ford-Fulkerson's Maximum Flow Algorithm,
Formalized Mathematics 13(2), pages 305-314, 2005. MML Identifier: GLIB_005

**Summary**: We define and prove correctness of Ford/Fulkerson's Maximum Network-Flow algorithm at the level of graph manipulations.

- Artur Kornilowicz.
Properties of Connected Subsets of the Real Line,
Formalized Mathematics 13(2), pages 315-323, 2005. MML Identifier: RCOMP_3

**Summary**: {}

- Artur Kornilowicz.
The Fundamental Group of the Circle,
Formalized Mathematics 13(2), pages 325-331, 2005. MML Identifier: TOPALG_5

**Summary**: {}

- Artur Kornilowicz, Yasunari Shidama.
Brouwer Fixed Point Theorem for Disks on the Plane,
Formalized Mathematics 13(2), pages 333-336, 2005. MML Identifier: BROUWER

**Summary**: {}

- Karol Pcak.
Stirling Numbers of the Second Kind,
Formalized Mathematics 13(2), pages 337-345, 2005. MML Identifier: STIRL2_1

**Summary**: In this paper we define Stirling numbers of the second kind by cardinality of certain functional classes such that $$S(n,k) = \{f {\rm ~where~} f {\rm~is~function~of~}n,k : f {\rm ~is~onto~"increasing} \}$$ After that we show basic properties of this number in order to prove recursive dependence of Stirling number of the second kind. Next theorems are introduced to prove formula $$S(n,k) = \frac{1}{k!} \mathop\Sigma_{i=0}^{k-1}(-1)^i {{k}\choose{i}}(k-i)^n$$ where $k\leq n.$

- Bo Zhang, Hiroshi Yamazaki, Yatsuka Nakamura.
Limit of Sequence of Subsets,
Formalized Mathematics 13(2), pages 347-352, 2005. MML Identifier: SETLIM_1

**Summary**: A concept of "limit of sequence of subsets" is defined here. This article contains the following items: 1. definition of the superior sequence and the inferior sequence of sets. 2. definition of the superior limit and the inferior limit of sets, and additional properties for the sigma-field of sets. and 3, definition of the limit value of a convergent sequence of sets, and additional properties for the sigma-field of sets.

- Magdalena Jastrzcebska, Adam Grabowski.
The Properties of Supercondensed Sets, Subcondensed Sets and Condensed Sets,
Formalized Mathematics 13(2), pages 353-359, 2005. MML Identifier: ISOMICHI

**Summary**: We formalized article ``New concepts in the theory of topological space -- supercondensed set, subcondensed set, and condensed set'' by Yoshinori Isomichi. First we defined supercondensed, subcondensed, and condensed sets and then gradually, defining other attributes such as regular open set or regular closed set, we formalized all the theorems and remarks that one can find in Isomichi's article.

- Yatsuka Nakamura, Andrzej Trybulec.
The Fashoda Meet Theorem for Rectangles,
Formalized Mathematics 13(2), pages 199-219, 2005. MML Identifier: JGRAPH_7
- Number 3
- Krzysztof Retel.
Properties of First and Second Order Cutting of Binary Relations,
Formalized Mathematics 13(3), pages 361-365, 2005. MML Identifier: RELSET_2

**Summary**: This paper introduces some notions concerning binary relations according to \cite{Riguet}. It is also an attempt to complement the knowledge contained in Mizar Mathematical Library regarding binary relations. We define here an image and inverse image of element of set A under binary relation of two sets A,B as image and inverse image of singleton of the element under this relation, respectively. Next, we define "The First Order Cutting Relation of two sets A,B under a subset of the set A" as the union of images of elements of this subset under the relation. We also define "The Second Order Cutting Subset of the Cartesian Product of two sets A,B under a subset of the set A" as an intersection of images of elements of this subset under the subset of the cartesian product. The paper also defines first and second projection of binary relations. The main goal of the article is to prove properties and collocations of introduced definitions in this paper. The numbers written in parenthesis after the label of theorems correspond to the numbers of expressions contained in the original article.

- Wenpai Chang, Hiroshi Yamazaki, Yatsuka Nakamura.
The Inner Product and Conjugate of Finite Sequences of Complex Numbers,
Formalized Mathematics 13(3), pages 367-373, 2005. MML Identifier: COMPLSP2

**Summary**: A concept of "the inner product and conjugate of finite sequences of complex numbers" is defined here. Addition, subtraction, Scalar multiplication and inner product are introduced using correspondent definitions of "conjugate of finite sequences of Field". Many equations for such operations consist like a case of "conjugate of finite sequences of Field". Some operations on the set of $n$-tuples of complex numbers are introduced as well. Addition, difference of such $n$-tuples, complement of a $n$-tuple and multiplication of these are defined in terms of complex numbers.

- Bo Zhang, Hiroshi Yamazaki, Yatsuka Nakamura.
Inferior Limit and Superior Limit of Sequences of Real Numbers,
Formalized Mathematics 13(3), pages 375-381, 2005. MML Identifier: RINFSUP1

**Summary**: A concept of inferior limit and superior limit of sequences of real numbers is defined here. This article contains the following items: definition of the superior sequence and the inferior sequence of real numbers, definition of the superior limit and the inferior limit of real number, and definition of the relation between the limit value and the superior limit, the inferior limit of sequences of real numbers.

- Fuguo Ge, Xiquan Liang, Yuzhong Ding.
Formulas and Identities of Inverse Hyperbolic Functions,
Formalized Mathematics 13(3), pages 383-387, 2005. MML Identifier: SIN_COS7

**Summary**: This article describes definitions of inverse hyperbolic functions and their main properties, as well as some addition formulas with hyperbolic functions.

- Akihiro Kubo.
Lines on Planes in $n$-Dimensional Euclidean Spaces,
Formalized Mathematics 13(3), pages 389-397, 2005. MML Identifier: EUCLIDLP

**Summary**: In this paper, we introduce basic properties of lines in the plane on this space. Lines and planes are expressed by the vector equation and are the image of $R$ and $R^2.$ By this, we can say that the properties of the classic Euclid geometry are satisfied also in $R^n$ as we know them intuitively. Next, we define the metric between the point and the line of this space.

- Karol Pcak.
Cardinal Numbers and Finite Sets,
Formalized Mathematics 13(3), pages 399-406, 2005. MML Identifier: CARD_FIN

**Summary**: In this paper we define class of functions and operators needed for proof of the principle of inclusions and the disconnections. We given also certain cardinal numbers concerning elementary class of functions (of course this function mapping finite set in finite set).

- Bo Zhang, Hiroshi Yamazaki, Yatsuka Nakamura.
Some Equations Related to the Limit of Sequence of Subsets,
Formalized Mathematics 13(3), pages 407-412, 2005. MML Identifier: SETLIM_2

**Summary**: Set operations for sequences of subsets are introduced here. Some relations for these operations with the limit of sequences of subsets, also with the inferior sequence and the superior sequence of sets, and with the inferior limit and the superior limit of sets are shown.

- Fuguo Ge, Xiquan Liang.
On the Partial Product of Series and Related Basic Inequalities,
Formalized Mathematics 13(3), pages 413-416, 2005. MML Identifier: SERIES_3

**Summary**: This article describes definition of partial product of series, introduced similarly to its related partial sum, as well as several important inequalities true for chosen special series.

- Masami Tanaka, Hiroshi Imura, Yatsuka Nakamura.
Homeomorphism between Finite Topological Spaces, Two-Dimensional Lattice Spaces and a Fixed Point Theorem,
Formalized Mathematics 13(3), pages 417-419, 2005. MML Identifier: FINTOPO5

**Summary**: In this paper, we first introduced the notion of homeomorphism between finite topological spaces. We also gave a fixed point theorem in finite topological space. Next, we showed two 2-dimensional concrete models of lattice spaces. One was 2-dimensional linear finite topological space. Another was 2-dimensional small finite topological space.

- Akira Nishino, Yasunari Shidama.
The Maclaurin Expansions,
Formalized Mathematics 13(3), pages 421-425, 2005. MML Identifier: TAYLOR_2

**Summary**: A concept of the Maclaurin expansions is defined here. This article contains the definition of the Maclaurin expansion and expansions of exp, sin and cos functions.

- Yan Zhang, Xiquan Liang.
Several Differentiable Formulas of Special Functions,
Formalized Mathematics 13(3), pages 427-434, 2005. MML Identifier: FDIFF_4

**Summary**: In this article, we give several differentiable formulas of special functions.

- Krzysztof Retel.
Properties of First and Second Order Cutting of Binary Relations,
Formalized Mathematics 13(3), pages 361-365, 2005. MML Identifier: RELSET_2
- Number 4
- Bo Zhang, Hiroshi Yamazaki, Yatsuka Nakamura.
Set Sequences and Monotone Class,
Formalized Mathematics 13(4), pages 435-441, 2005. MML Identifier: PROB_3

**Summary**: In this paper, we first defined the partial-union sequence, the partial-intersection sequence, and the partial-difference-union sequence of given sequence of subsets, and then proved the additive theorem of infinite sequences and sub-additive theorem of finite sequences for probability. Further, we defined the monotone class of families of subsets, and discussed about the relations between the monotone class and the $\sigma$-field which are generated by field of subsets of a given set.

- Hirofumi Fukura, Yatsuka Nakamura.
A Theory of Sequential Files,
Formalized Mathematics 13(4), pages 443-446, 2005. MML Identifier: FILEREC1

**Summary**: This article is a continuation of \cite{FINSEQ_8.ABS}. We present a notion of files and records. These are two finite sequences. One is a record and another is a separator for the carriage return and/or line feed. So, we define a record. The sequential text file contains records and separators. Generally, a record and a separator are paired in the file. And in a special situation, the separator does not exist in the file, for that the record is only one record or record is nothing. And the record does not exist in the file, for that some separator is in file. In this article, we present some theory for files and records.

- Fahui Zhai, Jianbing Cao, Xiquan Liang.
Circled Sets, Circled Hull, and Circled Family,
Formalized Mathematics 13(4), pages 447-451, 2005. MML Identifier: CIRCLED1

**Summary**: In this article, we prove some basic properties of the circled sets. We also define the circled hull, and give the definition of circled family.

- Adam Grabowski.
On the Borel Families of Subsets of Topological Spaces,
Formalized Mathematics 13(4), pages 453-461, 2005. MML Identifier: TOPGEN_4

**Summary**: This is the next Mizar article in a series aiming at complete formalization of ``General Topology'' \cite{ENGEL:1} by Engelking. We cover the second part of Section 1.3.

- Noboru Endou, Yasunari Shidama.
Linearity of Lebesgue Integral of Simple Valued Function,
Formalized Mathematics 13(4), pages 463-465, 2005. MML Identifier: MESFUNC4

**Summary**: In this article, the authors prove linearity of Lebesgue integral of simple valued function.

- Yatsuka Nakamura, Andrzej Trybulec, Artur Kornilowicz.
The Fashoda Meet Theorem for Continuous Mappings,
Formalized Mathematics 13(4), pages 467-469, 2005. MML Identifier: JGRAPH_8

**Summary**: {}

- Artur Kornilowicz, Grzegorz Bancerek, Adam Naumowicz.
Tietze Extension Theorem,
Formalized Mathematics 13(4), pages 471-475, 2005. MML Identifier: TIETZE

**Summary**: In this paper we formalize the Tietze extension theorem using as a basis the proof presented at the PlanetMath web server (\url{http://planetmath.org/encyclopedia/ProofOfTietzeExtensionTheorem2.html}).

- Adam Naumowicz, Grzegorz Bancerek.
Homeomorphisms of Jordan Curves,
Formalized Mathematics 13(4), pages 477-480, 2005. MML Identifier: JORDAN24

**Summary**: In this paper we prove that simple closed curves can be homeomorphically framed into a given rectangle. We also show that homeomorphisms preserve the Jordan property.

- Artur Kornilowicz.
Jordan Curve Theorem,
Formalized Mathematics 13(4), pages 481-491, 2005. MML Identifier: JORDAN

**Summary**: This paper formalizes the Jordan Curve Theorem following \cite{BrouwerJordan}.

- Wenpai Chang, Hiroshi Yamazaki, Yatsuka Nakamura.
The Inner Product and Conjugate of Matrix of Complex Numbers,
Formalized Mathematics 13(4), pages 493-499, 2005. MML Identifier: MATRIXC1

**Summary**: Concepts of the inner product and conjugate of matrix of complex numbers are defined here. Operations such as addition, subtraction, scalar multiplication and inner product are introduced using correspondent definitions of the conjugate of a matrix of a complex field. Many equations for such operations consist like a case of the conjugate of matrix of a field and some operations on the set of sum of complex numbers are introduced.

- Jianbing Cao, Fahui Zhai, Xiquan Liang.
Partial Sum and Partial Product of Some Series,
Formalized Mathematics 13(4), pages 501-503, 2005. MML Identifier: SERIES_4

**Summary**: This article contains partial sum and partial product of some series which are often used.

- Jianbing Cao, Fahui Zhai, Xiquan Liang.
Some Differentiable Formulas of Special Functions,
Formalized Mathematics 13(4), pages 505-509, 2005. MML Identifier: FDIFF_5

**Summary**: This article contains some differentiable formulas of special functions.

- Pacharapokin Chanapat, Hiroshi Yamazaki.
Formulas and Identities of Hyperbolic Functions,
Formalized Mathematics 13(4), pages 511-513, 2005. MML Identifier: SIN_COS8

**Summary**: In this article, we proved formulas of hyperbolic sine, hyperbolic cosine and hyperbolic tangent, and their identities.

- Grzegorz Bancerek.
Niemytzki Plane - an Example of Tychonoff Space Which Is Not $T_4$,
Formalized Mathematics 13(4), pages 515-524, 2005. MML Identifier: TOPGEN_5

**Summary**: We continue Mizar formalization of General Topology according to the book \cite{ENGEL:1} by Engelking. Niemytzki plane is defined as halfplane $y \geq 0$ with topology introduced by a neighborhood system. Niemytzki plane is not $T_4$. Next, the definition of Tychonoff space is given. The characterization of Tychonoff space by prebasis and the fact that Tychonoff spaces are between $T_3$ and $T_4$ is proved. The final result is that Niemytzki plane is also a Tychonoff space.

- Fuguo Ge, Xiquan Liang.
On the Partial Product and Partial Sum of Series and Related Basic Inequalities,
Formalized Mathematics 13(4), pages 525-528, 2005. MML Identifier: SERIES_5

**Summary**: This article introduced some important inequalities on partial sum and partial product, as well as some basic inequalities.

- Yan Zhang, Bo Li, Xiquan Liang.
Several Differentiable Formulas of Special Functions. Part II,
Formalized Mathematics 13(4), pages 529-535, 2005. MML Identifier: FDIFF_6

**Summary**: In this article, we give other several differentiable formulas of special functions.

- Shunichi Kobayashi.
On the Calculus of Binary Arithmetics. Part II,
Formalized Mathematics 13(4), pages 537-540, 2005. MML Identifier: BINARI_6

**Summary**:

- Xiaopeng Yue, Xiquan Liang, Zhongpin Sun.
Some Properties of Some Special Matrices,
Formalized Mathematics 13(4), pages 541-547, 2005. MML Identifier: MATRIX_6

**Summary**: This article describes definitions of reversible matrix, symmetrical matrix, antisymmetric matrix, orthogonal matrix and their main properties.

- Shin'nosuke Yamaguchi, Katsumi Wasaki, Nobuhiro Shimoi.
Generalized Full Adder Circuits (GFAs). Part I,
Formalized Mathematics 13(4), pages 549-571, 2005. MML Identifier: GFACIRC1

**Summary**: We formalize the concept of the Generalized Full Addition and Subtraction circuits (GFAs), define the structures of calculation units for the redundant signed digit (RSD) operations, and prove the stability of the circuits. Generally, one-bit binary full adder assumes positive weights to all of its three binary inputs and two outputs. We obtain four type of 1-bit GFA to constract the RSD arithmetic logical units that we generalize full adder to have both positive and negative weights to inputs and outputs.

- Artur Kornilowicz.
Quotient Rings,
Formalized Mathematics 13(4), pages 573-576, 2005. MML Identifier: RING_1

**Summary**: The notions of prime ideals and maximal ideals of a ring are introduced. Quotient rings are defined. Characterisation of of prime and maximal ideals using quotient rings are proved.

- Noboru Endou, Yasunari Shidama.
Completeness of the Real Euclidean Space,
Formalized Mathematics 13(4), pages 577-580, 2005. MML Identifier: REAL_NS1

**Summary**: {}

- Bo Zhang, Hiroshi Yamazaki, Yatsuka Nakamura.
Set Sequences and Monotone Class,
Formalized Mathematics 13(4), pages 435-441, 2005. MML Identifier: PROB_3
**Volume 14, 2006**- Number 1
- Yatsuka Nakamura.
Determinant of Some Matrices of Field Elements,
Formalized Mathematics 14(1), pages 1-5, 2006. MML Identifier: MATRIX_7

**Summary**: Here, we present determinants of some square matrices of field elements. First, the determinat of $2*2$ matrix is shown. Secondly, the determinants of zero matrix and unit matrix are shown, which are equal to 0 in the field and 1 in the field respectively. Thirdly, the determinant of diagonal matrix is shown, which is a product of all diagonal elements of the matrix. At the end, we prove that the determinant of a matrix is same as the determinant of its transpose.

- Xiaopeng Yue, Dahai Hu, Xiquan Liang.
Some Properties of Some Special Matrices. Part II,
Formalized Mathematics 14(1), pages 7-12, 2006. MML Identifier: MATRIX_8

**Summary**: This article describes definitions of Idempotent Matrix, Nilpotent Matrix, Involutory Matrix, Self Reversible Matrix, Similar Matrix, Congruent Matrix, the Trace of a Matrix and their main properties.

- Ewa Romanowicz, Adam Grabowski.
On the Permanent of a Matrix,
Formalized Mathematics 14(1), pages 13-20, 2006. MML Identifier: MATRIX_9

**Summary**: We introduce the notion of a permanent of a square matrix. It is a notion somewhat related to a determinant so we follow closely the approach and theorems already introduced in the Mizar Mathematical Library for the determinant. Unfortunately, the formalization of the latter notion is at its early stage, so we had to prove many very elementary auxiliary facts.

- Yatsuka Nakamura, Nobuyuki Tamura, Wenpai Chang.
A Theory of Matrices of Real Elements,
Formalized Mathematics 14(1), pages 21-28, 2006. MML Identifier: MATRIXR1

**Summary**: Here, the concept of matrix of real elements is introduced. This is defined as a special case of the general concept of matrix of a field. For such a real matrix, the notions of addition, subtraction, scalar product are defined. For any real finite sequences, two transformations to matrices are introduced. One of the matrices is of width 1, and the other is of length 1. By such transformations, two products of a matrix and a finite sequence are defined. The linearity of such product is shown.

- Magdalena Jastrzcebska, Adam Grabowski.
On the Properties of the M\"obius Function,
Formalized Mathematics 14(1), pages 29-36, 2006. MML Identifier: MOEBIUS1

**Summary**: We formalized some basic properties of the M\"obius function which is defined classically as $$\mu(n) = \begin{cases} 1, \textrm{~if~} n = 1,\\ 0, \textrm{~if~}p^2|n \textrm{~for~some~prime~}p\\ (-1)^r, \textrm{~if~}n=p_1 p_2 \cdots p_r, \textrm{~where~} p_i \textrm{~are~distinct~primes.}\\ \end{cases},$$ as e.g., its multiplicativity. To enable smooth reasoning about the sum of this number-theoretic function, we introduced an underlying manysorted set indexed by natural numbers. Its elements are just values of the M\"obius function.\par The second part of the paper is devoted to the notion of the radical of number, i.e. the product of its all prime factors.\par The formalization (which is very much like the one developed in Isabelle proof assistant connected with Avigad's formal proof of Prime Number Theorem) was done according to the book \cite{HardyWright}.

- Bo Li, Yan Zhang, Xiquan Liang.
Several Differentiation Formulas of Special Functions. Part III,
Formalized Mathematics 14(1), pages 37-45, 2006. MML Identifier: FDIFF_7

**Summary**: In this article, we give several differentiation formulas of special and composite functions including trigonometric function, inverse trigonometric function, polynomial function and logarithmic function.

- Yatsuka Nakamura.
Determinant of Some Matrices of Field Elements,
Formalized Mathematics 14(1), pages 1-5, 2006. MML Identifier: MATRIX_7
- Number 2
- Marco Riccardi.
Pocklington's Theorem and Bertrand's Postulate,
Formalized Mathematics 14(2), pages 47-52, 2006. MML Identifier: NAT_4

**Summary**: The first four sections include some auxiliary theorems related to number and finite sequence of numbers, in particular a primality test, the Pocklington's theorem (see \cite{LeVeque}). The last section presents the formalization of Bertrand's postulate closely following the book \cite{PFTB}, pp. 7--9.

- Noboru Endou, Yasunari Shidama.
Integral of Measurable Function,
Formalized Mathematics 14(2), pages 53-70, 2006. MML Identifier: MESFUNC5

**Summary**: In this paper we construct integral of measurable function.

- Marco Riccardi.
Pocklington's Theorem and Bertrand's Postulate,
Formalized Mathematics 14(2), pages 47-52, 2006. MML Identifier: NAT_4
- Number 3
- Bo Li, Yan Zhang, Artur Kornilowicz.
Simple Continued Fractions and Their Convergents,
Formalized Mathematics 14(3), pages 71-78, 2006. MML Identifier: REAL_3

**Summary**: The article introduces simple continued fractions. They are defined as an infinite sequence of integers. The characterization of rational numbers in terms of simple continued fractions is shown. We also give definitions of convergents of continued fractions, and several important properties of simple continued fractions and their convergents.

- Broderick Arneson, Piotr Rudnicki.
Chordal Graphs,
Formalized Mathematics 14(3), pages 79-92, 2006. MML Identifier: CHORD

**Summary**: We are formalizing \cite[pp.~81--84]{Golumbic} where chordal graphs are defined and their basic characterization is given. This formalization is a part of the M.Sc. work of the first author under supervision of the second author.

- Yatsuka Nakamura.
Connectedness and Continuous Sequences in Finite Topological Spaces,
Formalized Mathematics 14(3), pages 93-100, 2006. MML Identifier: FINTOPO6

**Summary**: First, equivalence conditions for connectedness are examined, for a finite topological space (originated in \cite{Nakamura:2}). Secondly, definitions of subspace, and components of the subspace of a finite topological space are given. Lastly, concepts of continuous finite sequence and minimum path of finite topological space are proposed.

- Bo Zhang, Yatsuka Nakamura.
The Definition of Finite Sequences and Matrices of Probability, and Addition of Matrices of Real Elements,
Formalized Mathematics 14(3), pages 101-108, 2006. MML Identifier: MATRPROB

**Summary**: In this article, we first define finite sequences of probability distribution and matrices of joint probability and conditional probability. We discuss also the concept of marginal probability. Further, we describe some theorems of matrices of real elements including quadratic form.

- Bo Li, Peng Wang.
Several Differentiation Formulas of Special Functions. Part IV,
Formalized Mathematics 14(3), pages 109-114, 2006. MML Identifier: FDIFF_8

**Summary**: In this article, we give several differentiation formulas of special and composite functions including trigonometric function, polynomial function and logarithmic function.

- Bo Li, Yan Zhang, Xiquan Liang.
Difference and Difference Quotient,
Formalized Mathematics 14(3), pages 115-119, 2006. MML Identifier: DIFF_1

**Summary**: In this article, we give the definitions of forward difference, backward difference, central difference and difference quotient, and some important properties of them.

- Bo Li, Yan Zhang, Artur Kornilowicz.
Simple Continued Fractions and Their Convergents,
Formalized Mathematics 14(3), pages 71-78, 2006. MML Identifier: REAL_3
- Number 4
- Krzysztof Treyderowski, Christoph Schwarzweller.
Multiplication of Polynomials using Discrete Fourier Transformation,
Formalized Mathematics 14(4), pages 121-128, 2006. MML Identifier: POLYNOM8

**Summary**: In this article we define the Discrete Fourier Transformation for univariate polynomials and show that multiplication of polynomials can be carried out by two Fourier Transformations with a vector multiplication inbetween. Our proof follows the standard one found in the literature and uses Vandermonde matrices, see e.g. \cite{ModernComputerAlgebra}.

- Xiquan Liang, Fuguo Ge, Xiaopeng Yue.
Some Special Matrices of Real Elements and Their Properties,
Formalized Mathematics 14(4), pages 129-134, 2006. MML Identifier: MATRIX10

**Summary**: This article describes definitions of positive matrix, negative matrix, nonpositive matrix, nonnegative matrix, nonzero matrix, module matrix of real elements and their main properties, and we also give the basic inequalities in matrices of real elements.

- Christoph Schwarzweller, Agnieszka Rowinska-Schwarzweller.
Schur's Theorem on the Stability of Networks,
Formalized Mathematics 14(4), pages 135-142, 2006. MML Identifier: HURWITZ

**Summary**: A complex polynomial is called a Hurwitz polynomial, if all its roots have a real part smaller than zero. This kind of polynomial plays an all-dominant role in stability checks of electrical networks.\par In this article we prove Schur's criterion \cite{SCHUR:1} that allows to decide whether a polynomial $p(x)$ is Hurwitz without explicitely computing its roots: Schur's recursive algorithm successively constructs polynomials $p_i(x)$ of lesser degree by division with $x-c$, $\Re\{c\} < 0$.

- Yasunari Shidama, Noboru Endou.
Integral of Real-Valued Measurable Function,
Formalized Mathematics 14(4), pages 143-152, 2006. MML Identifier: MESFUNC6

**Summary**: Based on \cite{Halmos}, authors formalized the integral of an extended real valued measurable function in \cite{MESFUNC5.ABS} before. However, the integral argued in \cite{MESFUNC5.ABS} cannot be applied to real-valued functions unconditionally. Therefore we formalized the integral of a real-value function in this article.

- Karol Pcak.
The Catalan Numbers. Part II,
Formalized Mathematics 14(4), pages 153-159, 2006. MML Identifier: CATALAN2

**Summary**: In this paper, we define sequence dominated by $0$, in which every initial fragment contains more zeroes than ones. If $n \geq 2 \cdot m$ and $n>0$, then the number of sequences dominated by $0$ the length $n$ including $m$ of ones, is given by the formula $$D(n,m)=\frac{n+1-2\cdot m}{n+1-m}\cdot{n \choose m}$$ and satisfies the recurrence relation $$D(n,m)=D(n-1,m)\:+\:\sum_{i=0}^{m-1} D(2\cdot i,i)\cdot D(n-2\cdot(i+1),m - (i+1)).$$ Obviously, if $n = 2 \cdot m$, then we obtain the recurrence relation for the Catalan numbers (starting from $0$) $$C_{m+1}=\sum_{i=0}^{m-1} C_{i+1}\cdot C_{m-i}.$$ Using the above recurrence relation we can see that $$\sum_{i=0}^\infty\,C_{i+1}\cdot x^i\,= \,1\,+\,\left(\sum_{i=0}^\infty\, C_{i+1}\cdot x^i\right)^2$$ where ($| x |< \frac{1}{4}$) and hence $$\sum_{i=0}^\infty\, C_{i+1}\cdot x^i\,= \,\frac{1-\sqrt[]{1-4\cdot x}}{2\cdot x}.$$

- Xiquan Liang, Fuguo Ge.
The Quaternion Numbers,
Formalized Mathematics 14(4), pages 161-169, 2006. MML Identifier: QUATERNI

**Summary**: In this article, we define the set $\Bbb Q$ of quaternion numbers as the set of all ordered sequences $q =\langle x,y,w,z\rangle$ where $x$,$y$,$w$ and $z$ are real numbers. The addition, difference and multiplication of the quaternion numbers are also defined. We define the real and imaginary parts of $q$ and denote this by $x = \Rea(q)$, $y = \Im1(q)$, $w = \Im2(q)$, $z = \Im3(q)$. We define the addition, difference, multiplication again and denote this operation by real and three imaginary parts. We define the conjugate of $q$ denoted by $q*'$ and the absolute value of $q$ denoted by $|.q.|$. We also give some properties of quaternion numbers.

- Kazuhisa Ishida.
Model Checking. Part I,
Formalized Mathematics 14(4), pages 171-186, 2006. MML Identifier: MODELC_1

**Summary**: This text includes definitions of the Kripke structure, CTL (Computation Tree Logic), and verification of the basic algorithm for Model Checking based on CTL. Text book for reference: Model Checking, E. M. Clarke, Orna Grumberg, Doron Peled, Mit Pr (2000/1/7)

- Broderick Arneson, Piotr Rudnicki.
Recognizing Chordal Graphs: Lex BFS and MCS,
Formalized Mathematics 14(4), pages 187-205, 2006. MML Identifier: LEXBFS

**Summary**: We are formalizing the algorithm for recognizing chordal graphs by lexicographic breadth-first search as presented in \cite[Section 3 of Chapter4, pp.~81--84]{Golumbic}. Then we follow with a formalization of another algorithm serving the same end but based on maximum cardinality search as presented by Tarjan and Yannakakis~\cite{TY84}.\par This work is a part of the MSc work of the first author under supervision of the second author. We would like to thank one of the anonymous reviewers for very useful suggestions.

- Noboru Endou, Yasunari Shidama, Masahiko Yamazaki.
Integrability and the Integral of Partial Functions from $\Bbb R$ into $\Bbb R$,
Formalized Mathematics 14(4), pages 207-212, 2006. MML Identifier: INTEGRA6

**Summary**:

- Noboru Endou, Yasunari Shidama, Katsumasa Okamura.
Baire's Category Theorem and Some Spaces Generated from Real Normed Space,
Formalized Mathematics 14(4), pages 213-219, 2006. MML Identifier: NORMSP_2

**Summary**: As application of complete metric space, we proved a Baire's category theorem. Then we defined some spaces generated from real normed space and discussed about each. In the second section we showed an equivalence of convergence and a continuity of a function. In other sections, we showed some topological properties of two spaces, which are topological space and linear topological space generated from real normed space.

- Adam Naumowicz.
On the Representation of Natural Numbers in Positional Numeral Systems,
Formalized Mathematics 14(4), pages 221-223, 2006. MML Identifier: NUMERAL1

**Summary**: In this paper we show that every natural number can be uniquely represented as a base-$b$ numeral. The formalization is based on the proof presented in \cite{SIERPINSKI:1}. We also prove selected divisibility criteria in the base-$10$ numeral system.

- Bo Zhang, Hiroshi Yamazaki and Yatsuka Nakamura.
The Relevance of Measure and Probability, and Definition of Completeness of Probability,
Formalized Mathematics 14(4), pages 225-229, 2006. MML Identifier: PROB_4

**Summary**: In this article, we first discuss the relation between measure defined using extended real numbers and probability defined using real numbers. Further, we define completeness of probability, and its completion method, and also show that they coincide with those of measure.

- Krzysztof Treyderowski, Christoph Schwarzweller.
Multiplication of Polynomials using Discrete Fourier Transformation,
Formalized Mathematics 14(4), pages 121-128, 2006. MML Identifier: POLYNOM8
**Volume 15, 2007**- Number 1
- Yuzhong Ding.
Several Classes of {BCI}-algebras
and their Properties,
Formalized Mathematics 15(1),
pages 1-9, 2007. MML Identifier: BCIALG_1

**Summary**: I have formalized the BCI-algebras closely following the book cite{BCIAlgebras}, sections 1.1 to 1.3, 1.6, 2.1 to 2.3, and 2.7. In this article the general theory of {BCI}-algebras and several classes of {BCI}-algebras are given.

- Micha{\l} Trybulec.
Formal Languages -- Concatenation and Closure,
Formalized Mathematics 15(1),
pages 11-15, 2007. MML Identifier: FLANG_1

**Summary**: Formal languages are introduced as subsets of the set of all 0-based finite sequences over a given set (the alphabet). Concatenation, the $n$-th power and closure are defined and their properties are shown. Finally, it is shown that the closure of the alphabet (understood here as the language of words of length 1) equals to the set of all words over that alphabet, and that the alphabet is the minimal set with this property. Notation and terminology were taken from \cite{HOPCROFT-ULLMAN:1979} and \cite{WAITE-GOOS:1984}.

- Karol P\c{a}k.
Basic Properties of Determinants
of Square Matrices over a Field,
Formalized Mathematics 15(1),
pages 17-25, 2007. MML Identifier: MATRIX11

**Summary**: In this paper I present basic properties of the determinant of square matrices over a field and selected properties of the sign of a permutation. First, I define the sign of a permutation by the requirement $${\rm sgn}(p)=\prod_{1\leq i \gt j \leq n}\,{\rm sgn}(p(j)-p(i)),$$ where $p$ is any fixed permutation of a set with $n$ elements. I prove that the sign of a product of two permutations is the same as the product of their signs and show the relation between signs and parity of permutations. Then I consider the determinant of a linear combination of lines, the determinant of a matrix with permutated lines and the determinant of a matrix with a repeated line. Finally, at the end I prove that the determinant of a product of two square matrices is equal to the product of their determinants.

- Yuzhong Ding.
Several Classes of {BCI}-algebras
and their Properties,
Formalized Mathematics 15(1),
pages 1-9, 2007. MML Identifier: BCIALG_1
- Number 2
- Andrzej Owsiejczuk.
Combinatorial Grassmannians,
Formalized Mathematics 15(2),
pages 27-33, 2007. MML Identifier: COMBGRAS

**Summary**: In the paper I construct the configuration $G$ which is a partial linear space. It consists of $k$-element subsets of some base set as points and $(k+1)$-element subsets as lines. The incidence is given by inclusion. I also introduce automorphisms of partial linear spaces and show that automorphisms of $G$ are generated by permutations of the base set.

- Marco Riccardi.
The Jordan-H\"older Theorem,
Formalized Mathematics 15(2),
pages 35-51, 2007. MML Identifier: GROUP_9

**Summary**: The goal of this article is to formalize the Jordan-H\"older theorem in the context of group with operators as in the book \cite{BourbakiAlgI}. Accordingly, the article introduces the structure of group with operators and reformulates some theorems on a group already present in the Mizar Mathematical Library. Next, the article formalizes the Zassenhaus butterfly lemma and the Schreier refinement theorem, and defines the composition series.

- Micha{\l} Trybulec.
Regular Expression Quantifiers -- m to n Occurrences,
Formalized Mathematics 15(2),
pages 53-58, 2007. MML Identifier: FLANG_2

**Summary**: This article includes proofs of several facts that are supplemental to the theorems proved in \cite{FLANG_1.ABS}. Next, it builds upon that theory to extend the framework for proving facts about formal languages in general and regular expression operators in particular. In this article, two quantifiers are defined and their properties are shown: $m$ to $n$ occurrences (or the union of a range of powers) and optional occurrence. Although optional occurrence is a special case of the previous operator (0 to 1 occurrences), it is often defined in regex applications as a separate operator -- hence its explicit definition and properties in the article. Notation and terminology were taken from \cite{WALL-CHRISTIANSEN-ORWANT:2000}.

- Yasunari Shidama, Noboru Endou, Katsumi Wasaki.
Riemann Indefinite Integral of Functions
of Real Variable,
Formalized Mathematics 15(2),
pages 59-63, 2007. MML Identifier: INTEGRA7

**Summary**: In this article we define the Riemann indefinite integral of functions of real variable and prove the linearity of that \cite{Apostol:1969}. And we give some examples of the indefinite integral of some elementary functions. Furthermore, also the theorem about integral operation and uniform convergent sequence of functions is proved.

- Noboru Endou, Yasunari Shidama, Keiichi Miyajima.
Partial Differentiation on Normed Linear
Spaces ${\cal R}^n$,
Formalized Mathematics 15(2),
pages 65-72, 2007. MML Identifier: PDIFF_1

**Summary**: In this article, we define the partial differentiation of functions of real variable and prove the linearity of this operator \cite{Schwartz:1981}.

- Andrzej Owsiejczuk.
Combinatorial Grassmannians,
Formalized Mathematics 15(2),
pages 27-33, 2007. MML Identifier: COMBGRAS
- Number 3
- Peng Wang, Bo Li.
Several Differentiation Formulas of Special Functions. {P}art {V},
Formalized Mathematics 15(3),
pages 73-79, 2007. MML Identifier: FDIFF_9

**Summary**: In this article, we give several differentiation formulas of special and composite functions including trigonometric, polynomial and logarithmic functions.

- Noboru Endou, Yasunari Shidama, Keiichi Miyajima.
The Product Space of Real Normed Spaces and its Properties,
Formalized Mathematics 15(3),
pages 81-85, 2007. MML Identifier: PRVECT_2

**Summary**: In this article, we define the product space of real linear spaces and real normed spaces. We also describe properties of these spaces.

- Grzegorz Bancerek.
Mizar Analysis of Algorithms: {P}reliminaries,
Formalized Mathematics 15(3),
pages 87-110, 2007. MML Identifier: AOFA_000

**Summary**: Algorithms and its parts -- instructions -- are formalized as elements of if-while algebras. An if-while algebra is a (1-sorted) universal algebra which has 4 operations: a constant -- the empty instruction, a binary catenation of instructions, a ternary conditional instruction, and a binary while instruction. An execution function is defined on pairs $(s,I)$, where $s$ is a state (an element of certain set of states) and $I$ is an instruction, and results in states. The execution function obeys control structures using the set of distinguished true states, i.e. a condition instruction is executed and the continuation of execution depends on if the resulting state is in true states or not. Termination is also defined for pairs $(s,I)$ and depends on the execution function. The existence of execution function determined on elementary instructions and its uniqueness for terminating instructions are shown.

- Bo Zhang, Yatsuka Nakamura.
Definition and some Properties of Information Entropy,
Formalized Mathematics 15(3),
pages 111-119, 2007. MML Identifier: ENTROPY1

**Summary**: In this article we mainly define the information entropy \cite{Billingsley:1964}, \cite{Hirasawa:1996} and prove some its basic properties. First, we discuss some properties on four kinds of transformation functions between vector and matrix. The transformation functions are LineVec2Mx, ColVec2Mx, Vec2DiagMx and Mx2FinS. Mx2FinS is a horizontal concatenation operator for a given matrix, treating rows of the given matrix as finite sequences, yielding a new finite sequence by horizontally joining each row of the given matrix in order to index. Then we define each concept of information entropy for a probability sequence and two kinds of probability matrices, joint and conditional, that are defined in article \cite{MATRPROB.ABS}. Further, we discuss some properties of information entropy including Shannon's lemma, maximum property, additivity and super-additivity properties.

- Micha{\l} ~Trybulec.
String Rewriting Systems,
Formalized Mathematics 15(3),
pages 121-126, 2007. MML Identifier: REWRITE2

**Summary**: Basing on the definitions from \cite{WAITE-GOOS:1984}, semi-Thue systems, Thue systems, and direct derivations are introduced. Next, the standard reduction relation is defined that, in turn, is used to introduce derivations using the theory from \cite{REWRITE1.ABS}. Finally, languages generated by rewriting systems are defined as all strings reachable from an initial word. This is followed by the introduction of the equivalence of semi-Thue systems with respect to the initial word.

- Nobuyuki Tamura, Yatsuka Nakamura.
Determinant and Inverse of Matrices of Real Elements,
Formalized Mathematics 15(3),
pages 127-136, 2007. MML Identifier: MATRIXR2

**Summary**: In this paper the classic theory of matrices of real elements (see e.g. \cite{Furuya:57}, \cite{Gantmacher:59}) is developed. We prove selected equations that have been proved previously for matrices of field elements. Similarly, we introduce in this special context the determinant of a matrix, the identity and zero matrices, and the inverse matrix. The new concept discussed in the case of matrices of real numbers is the property of matrices as operators acting on finite sequences of real numbers from both sides. The relations of invertibility of matrices and the ``onto" property of matrices as operators are discussed.

- Jesse ~Alama.
The Rank+Nullity Theorem,
Formalized Mathematics 15(3),
pages 137-142, 2007. MML Identifier: RANKNULL

**Summary**: The rank+nullity theorem states that, if $T$ is a linear transformation from a finite-dimensional vector space $V$ to a finite-dimensional vector space $W$, then $\dim(V) = \mathrm{rank}(T) + \mathrm{nullity}(T)$, where $\mathrm{rank}(T) = \dim(\mathrm{im}(T))$ and $\mathrm{nullity}(T) = \dim(\mathrm{ker}(T))$. The proof treated here is standard; see, for example,~\cite{lang-algebra}: take a basis $A$ of $\mathrm{ker}(T)$ and extend it to a basis $B$ of $V$, and then show that $\dim(\mathrm{im}(T))$ is equal to $|B - A|$, and that $T$ is one-to-one on $B - A$.

- Karol P\c{a}k, Andrzej Trybulec.
Laplace Expansion,
Formalized Mathematics 15(3),
pages 143-150, 2007. MML Identifier: LAPLACE

**Summary**: In the article the formula for Laplace expansion is proved.

- Xiquan Liang, Tao Sun, Dahai Hu.
Some Properties of Line and Column Operations on Matrices,
Formalized Mathematics 15(3),
pages 151-157, 2007. MML Identifier: MATRIX12

**Summary**: This article describes definitions of elementary operations about matrix and their main properties.

- Marco Riccardi.
The {S}ylow Theorems,
Formalized Mathematics 15(3),
pages 159-165, 2007. MML Identifier: GROUP_10

**Summary**: The goal of this article is to formalize the Sylow theorems closely following the book \cite{BourbakiAlgI}. Accordingly, the article introduces the group operating on a set, the stabilizer, the orbits, the $p$-groups and the Sylow subgroups.

- Peng Wang, Bo Li.
Several Differentiation Formulas of Special Functions. {P}art {V},
Formalized Mathematics 15(3),
pages 73-79, 2007. MML Identifier: FDIFF_9
- Number 4
- Czes{\l}aw Byli{\'n}ski.
Alexandroff One Point Compactification,
Formalized Mathematics 15(4),
pages 167-170, 2007. MML Identifier: COMPACT1

**Summary**: In the article, I introduce the notions of the compactification of topological spaces and the Alexandroff one point compactification. Some properties of the locally compact spaces and one point compactification are proved.

- Freek Wiedijk.
Arrow's Impossibility Theorem,
Formalized Mathematics 15(4),
pages 171-174, 2007. MML Identifier: ARROW

**Summary**: A formalization of the first proof from \cite{Geanakoplos:1996}.

- Yuzhong Ding, Zhiyong Pang.
Congruences and Quotient Algebras of {BCI}-algebras,
Formalized Mathematics 15(4),
pages 175-180, 2007. MML Identifier: BCIALG_2

**Summary**: We have formalized the BCI-algebras closely following the book \cite{BCIAlgebras2} pp.16-19 and pp.58-65. Firstly, the article focuses on the properties of the element and then the definition and properties of congruences and quotient algebras are given. Quotient algebras are the basic tools for exploring the structures of {BCI}-algebras.

- Xiquan Liang, Li Yan, Junjie Zhao.
Linear Congruence Relation and Complete Residue Systems,
Formalized Mathematics 15(4),
pages 181-187, 2007. MML Identifier: INT_4

**Summary**: In this paper, we defined the congruence relation and proved its fundamental properties on the base of some useful theorems. Then we proved the existence of solution and the number of incongruent solution to a linear congruence and the linear congruent equation class, in particular, we proved the Chinese Remainder Theorem. Finally, we defined the complete residue system and proved its fundamental properties.

- Cuiying Peng, Fuguo Ge, Xiquan Liang.
Several Integrability Formulas of Special Functions,
Formalized Mathematics 15(4),
pages 189-198, 2007. MML Identifier: INTEGRA8

**Summary**: In this article, we give several integrability formulas of special and composite functions including trigonometric function, inverse trigonometric function, hyperbolic function and logarithmic function.

- Karol P\c{a}k.
Basic Properties of the Rank of Matrices over a Field,
Formalized Mathematics 15(4),
pages 199-211, 2007. MML Identifier: MATRIX13

**Summary**: In this paper I present selected properties of triangular matrices and basic properties of the rank of matrices over a field.\par I define a submatrix as a matrix formed by selecting certain rows and columns from a bigger matrix. That is in my considerations, as an array, it is cut down to those entries constrained by row and column. Then I introduce the concept of the rank of a $m\times n$ matrix $A$ by the condition: $A$ has the rank $r$ if and only if, there is a $r\times r$ submatrix of $A$ with a non-zero determinant, and for every $k\times k$ submatrix of $A$ with a non-zero determinant we have $k\leq r$.\par At the end, I prove that the rank defined by the size of the biggest submatrix with a non-zero determinant of a matrix $A$, is the same as the maximal number of linearly independent rows of $A$.

- Klaus E. Grue, Artur Korni{\l}owicz.
Basic Operations on Preordered Coherent Spaces,
Formalized Mathematics 15(4),
pages 213-230, 2007. MML Identifier: PCS_0

**Summary**: This Mizar paper presents the definition of a ``Preordered Coherent Space" (PCS). Furthermore, the paper defines a number of operations on PCS's and states and proves a number of elementary lemmas about these operations. PCS's have many useful properties which could qualify them for mathematical study in their own right. PCS's were invented, however, to construct Scott domains, to solve domain equations, and to construct models of various versions of lambda calculus. For more on PCS's, see \cite{Krivine93}. The present Mizar paper defines the operations on PCS's used in Chapter 8 of \cite{berline97kappadenotational}.

- Hiroshi Yamazaki, Noboru Endou, Yasunari Shidama, Hiroyuki Okazaki.
Inferior Limit, Superior Limit and Convergence of Sequences of Extended Real Numbers,
Formalized Mathematics 15(4),
pages 231-235, 2007. MML Identifier: RINFSUP2

**Summary**: In this article, we extended properties of sequences of real numbers to sequences of extended real numbers. We also introduced basic properties of the inferior limit, superior limit and convergence of sequences of extended real numbers.

- Tao Sun, Dahai Hu, Xiquan Liang.
Several Classes of {BCK}-algebras and their Properties,
Formalized Mathematics 15(4),
pages 237-242, 2007. MML Identifier: BCIALG_3

**Summary**: In this article the general theory of Commutative BCK-algebras and BCI-algebras and several classes of {BCK}-algebras are given according to \cite{BCIAlgebras}.

- Bo Li, Pan Wang.
Several Differentiation Formulas of Special Functions. {P}art {VI},
Formalized Mathematics 15(4),
pages 243-250, 2007. MML Identifier: FDIFF_10

**Summary**: In this article, we prove a series of differentiation identities \cite{Chemnitius:1956} involving the secant and cosecant functions and specific combinations of special functions including trigonometric, exponential and logarithmic functions.

- Czes{\l}aw Byli{\'n}ski.
Alexandroff One Point Compactification,
Formalized Mathematics 15(4),
pages 167-170, 2007. MML Identifier: COMPACT1
**Volume 16, 2008**- Number 1
- Jesse Alama.
The Vector Space of Subsets of a
Set Based on Symmetric Difference,
Formalized Mathematics 16(1),
pages 1-5, 2008. MML Identifier:
BSPACE

**Summary**: For each set $X$, the power set of $X$ forms a vector space over the field ${\bf Z}_{2}$ (the two-element field $\{0,1\}$ with addition and multiplication done modulo $2$): vector addition is disjoint union, and scalar multiplication is defined by the two equations ($1 \cdot x := x$, $0 \cdot x := \emptyset$ for subsets $x$ of $X$). See \cite{kelley}, Exercise 2.K, for more information.

- Jesse Alama.
Euler's Polyhedron Formula,
Formalized Mathematics 16(1),
pages 7-17, 2008. MML Identifier:
POLYFORM

**Summary**: Euler's polyhedron theorem states for a polyhedron $p$, that $$ V - E + F = 2, $$ where $V$, $E$, and $F$ are, respectively, the number of vertices, edges, and faces of $p$. The formula was first stated in print by Euler in 1758 \cite{euler1758a}. The proof given here is based on Poincar\'e's linear algebraic proof, stated in \cite{poincare1893} (with a corrected proof in \cite{poincare1899}), as adapted by Imre Lakatos in the latter's \textsl{Proofs and Refutations} \cite{proofs-and-refutations}. \par As is well known, Euler's formula is not true for all polyhedra. The condition on polyhedra considered here is that of being a homology sphere, which says that the cycles (chains whose boundary is zero) are exactly the bounding chains (chains that are the boundary of a chain of one higher dimension).\par The present proof actually goes beyond the three-dimensional version of the polyhedral formula given by Lakatos; it is dimension-free, in the sense that it gives a formula in which the dimension of the polyhedron is a parameter. The classical Euler relation $V - E + F = 2$ is corresponds to the case where the dimension of the polyhedron is $3$. \par The main theorem, expressed in the language of the present article, is $$\mathtt{Sum \ alternating-characteristic-sequence(p) = 0},$$ where $p$ is a polyhedron. The alternating characteristic sequence of a polyhedron is the sequence $$ -N(-1), +N(0), -N(1), \dots, (-1)^{\dim(p)}*N(\dim(p)),$$ where $N(k)$ is the number of polytopes of $p$ of dimension $k$. The special case of $\dim(p) = 3$ yields Euler's classical relation. ($N(-1)$ and $N(3)$ will turn out to be equal, by definition, to $1$.) \par Two other special cases are proved: the first says that a one-dimensional ``polyhedron" that is a homology sphere consists of just two vertices (and thus consists of just a single edge); the second special case asserts that a two-dimensional polyhedron that is a homology sphere (a polygon) has as many vertices as edges. \par A treatment of the more general version of Euler's relation can be found in \cite{grunbaum2003} and \cite{brondsted1983}. The former contains a proof of Steinitz's theorem, which shows that the abstract polyhedra treated in Poincar\'e's proof, which might not appear to be about polyhedra in the usual sense of the word, are in fact embeddable in $\mathbf{R}^3$ under certain conditions. It would be valuable to formalize a proof of Steinitz's theorem and relate it to the development contained here.

- Hideki Sakurai, Hisayoshi Kunimune, Yasunari Shidama.
Uniform Boundedness Principle,
Formalized Mathematics 16(1),
pages 19-21, 2008. MML Identifier:
LOPBAN_5

**Summary**: In this article at first, we proved the lemma of the inferior limit and the superior limit. Next, we proved the Baire category theorem (Banach space version) \cite{yoshida:1980}, \cite{miyadera:1972}, \cite{Dunford:1958}, quoted it and proved the uniform boundedness principle. Moreover, the proof of the Banach-Steinhaus theorem is added.

- Li Yan, Xiquan Liang, Junjie Zhao.
Gauss Lemma and Law of Quadratic Reciprocity,
Formalized Mathematics 16(1),
pages 23-28, 2008. MML Identifier:
INT_5

**Summary**: In this paper, we defined the quadratic residue and proved its fundamental properties on the base of some useful theorems. Then we defined the Legendre symbol and proved its useful theorems \cite{HuaLooKeng:1957}, \cite{Dexin:1965}. Finally, Gauss Lemma and Law of Quadratic Reciprocity are proven.

- Micha{\l} Trybulec.
Regular Expression Quantifiers -- at least $m$ Occurrences,
Formalized Mathematics 16(1),
pages 29-33, 2008. MML Identifier:
FLANG_3

**Summary**: This is the second article on regular expression quantifiers. \cite{FLANG_2.ABS} introduced the quantifiers $m$ to $n$ occurrences and optional occurrence. In the sequel, the quantifiers: at least $m$ occurrences and positive closure (at least 1 occurrence) are introduced. Notation and terminology were taken from \cite{WALL-CHRISTIANSEN-ORWANT:2000}, several properties of regular expressions from \cite{WAITE-GOOS:1984}.

- Karol P\k{a}k.
Complete Spaces,
Formalized Mathematics 16(1),
pages 35-43, 2008. MML Identifier:
COMPL_SP

**Summary**: This paper is a continuation of \cite{TBSP_1.ABS}. First some definitions needed to formulate Cantor's theorem on complete spaces and show several facts about them are introduced. Next section contains the proof of Cantor's theorem and some properties of complete spaces resulting from this theorem. Moreover, countable compact spaces and proofs of auxiliary facts about them is defined. I also show the important condition that every metric space is compact if and only if it is countably compact. Then I prove that every metric space is compact if and only if it is a complete and totally bounded space. I also introduce the definition of the metric space with the well metric. This article is based on \cite{ENGEL:1}.

- Bo Li, Yanping Zhuang, Xiquan Liang.
Difference and Difference Quotient. {P}art {II},
Formalized Mathematics 16(1),
pages 45-49, 2008. MML Identifier:
DIFF_2

**Summary**: In this article, we give some important properties of forward difference, backward difference, central difference and difference quotient and forward difference, backward difference, central difference and difference quotient formulas of some special functions \cite{Renhong:1999}.

- Keiko Narita, Noboru Endou, Yasunari Shidama.
The First Mean Value Theorem for Integrals,
Formalized Mathematics 16(1),
pages 51-55, 2008. MML Identifier:
MESFUNC7

**Summary**: In this article, we prove the first mean value theorem for integrals \cite{Halmos74}. The formalization of various theorems about the properties of the Lebesgue integral is also presented.

- Noboru Endou, Yasunari Shidama, Keiko Narita.
Egoroff's Theorem,
Formalized Mathematics 16(1),
pages 57-63, 2008. MML Identifier:
MESFUNC8

**Summary**: The goal of this article is to prove Egoroff's Theorem \cite{Halmos74}. However, there are not enough theorems related to sequence of measurable functions in Mizar Mathematical Library. So we proved many theorems about them. At the end of this article, we showed Egoroff's theorem.

- Tao Sun, Junjie Zhao, Xiquan Liang.
BCI-algebras with Condition ({S})
and their Properties,
Formalized Mathematics 16(1),
pages 65-71, 2008. MML Identifier:
BCIALG_4

**Summary**: In this article we will first investigate the elementary properties of BCI-algebras with condition (S), see \cite{BCIAlgebras}. And then we will discuss the three classes of algebras: commutative, positive-implicative and implicative BCK-algebras with condition (S).

- Katsumi Wasaki.
Stability of $n$-Bit Generalized Full
Adder Circuits ({GFA}s). {P}art {II},
Formalized Mathematics 16(1),
pages 73-80, 2008. MML Identifier:
GFACIRC2

**Summary**: We continue to formalize the concept of the Generalized Full Addition and Subtraction circuits (GFAs), define the structures of calculation units for the Redundant Signed Digit (RSD) operations, then prove its stability of the calculations. Generally, one-bit binary full adder assumes positive weights to all of its three binary inputs and two outputs. We define the circuit structure of two-types $n$-bit GFAs using the recursive construction to use the RSD arithmetic logical units that we generalize full adder to have both positive and negative weights to inputs and outputs. The motivation for this research is to establish a technique based on formalized mathematics and its applications for calculation circuits with high reliability.

- Karol P\k{a}k.
Solutions of Linear Equations,
Formalized Mathematics 16(1),
pages 81-90, 2008. MML Identifier:
MATRIX15

**Summary**: In this paper I present the Kronecker-Capelli theorem which states that a system of linear equations has a solution if and only if the rank of its coefficient matrix is equal to the rank of its augmented matrix.

- Jesse Alama.
The Vector Space of Subsets of a
Set Based on Symmetric Difference,
Formalized Mathematics 16(1),
pages 1-5, 2008. MML Identifier:
BSPACE
- Number 2
- Jessica Enright, Piotr Rudnicki.
Helly Property for Subtrees,
Formalized Mathematics 16(2),
pages 91-96, 2008. MML Identifier:
HELLY

**Summary**: We prove, following \cite[p.92]{Golumbic}, that any family of subtrees of a finite tree satisfies the Helly property.

- Marco Riccardi.
Heron's Formula and {P}tolemy's Theorem,
Formalized Mathematics 16(2),
pages 97-101, 2008. MML Identifier:
EUCLID_6

**Summary**: The goal of this article is to formalize some theorems that are in the \cite{Freek-100-theorems} on the web. These are elementary theorems included in every handbook of Euclidean geometry and trigonometry: the law of cosines, the Heron's formula, the isosceles triangle theorem, the intersecting chords theorem and the Ptolemy's theorem.

- Hiroyuki Okazaki, Yasunari Shidama.
Uniqueness of Factoring
an Integer and Multiplicative Group ${\mathbb Z}/p{{\mathbb Z}^\ast}$,
Formalized Mathematics 16(2),
pages 103-107, 2008. MML Identifier:
INT_7

**Summary**: In the \cite{INT_3.ABS}, it had been proven that the Integers modulo $p$, in this article we shall refer as ${\mathbb Z}/p{\mathbb Z}$, constitutes a field if and only if $p$ is a prime. Then the prime modulo ${\mathbb Z}/p{\mathbb Z}$ is an additive cyclic group and ${\mathbb Z}/p{{\mathbb Z}^\ast}={\mathbb Z}/p{\mathbb Z}\setminus\{0\}$ is a multiplicative cyclic group, too. The former has been proven in the \cite{GR_CY_1.ABS}. However, the latter had not been proven yet. In this article, first, we prove a theorem concerning the LCM to prove the existence of primitive elements of ${\mathbb Z}/p^{\ast}$. Moreover we prove the uniqueness of factoring an integer. Next we define the multiplicative group ${\mathbb Z}/p{{\mathbb Z}^\ast}$ and prove it is cyclic.

- Chenglong Wu, Yuzhong Ding.
Ideals of {BCI}-algebras and their Properties,
Formalized Mathematics 16(2),
pages 109-114, 2008. MML Identifier:
BCIIDEAL

**Summary**: In this article three classes of ideals are discussed: associative ideals, commutative ideals, implicative ideals and positive implicative ideals, and their elementary properties. Some of their properties and the relationships between them have not been proven yet, and will be completed in the following article.

- Yasunari Shidama, Hikofumi Suzuki, Noboru Endou.
Banach Algebra of Bounded Functionals,
Formalized Mathematics 16(2),
pages 115-122, 2008. MML Identifier:
C0SP1

**Summary**: In this article, we describe some basic properties of the Banach algebra which is constructed from all bounded functionals.

- Hidenori Matsuzaki, Noboru Endou, Yasunari Shidama.
Convex Sets and Convex Combinations
on Complex Linear Spaces,
Formalized Mathematics 16(2),
pages 123-133, 2008. MML Identifier:
CONVEX4

**Summary**: In this article, convex sets, convex combinations and convex hulls on complex linear spaces are introduced.

- Fuguo Ge.
Inner Products, Group, Ring
of Quaternion Numbers,
Formalized Mathematics 16(2),
pages 135-139, 2008. MML Identifier:
QUATERN2

**Summary**: In this article, we define the division of the quaternion numbers, we also give the definition of inner products, group, ring of the quaternion numbers, and we prove some of their properties.

- Junjie Zhao, Xiquan Liang, Li Yan.
Several Higher Differentiation Formulas
of Special Functions,
Formalized Mathematics 16(2),
pages 141-145, 2008. MML Identifier:
HFDIFF_1

**Summary**: In this paper, we proved some basic properties of higher differentiation, and higher differentiation formulas of special functions \cite{Chen:1978}.

- Xiquan Liang, Bing Xie.
Inverse Trigonometric Functions
Arctan and Arccot,
Formalized Mathematics 16(2),
pages 147-158, 2008. MML Identifier:
SIN_COS9

**Summary**: This article describes definitions of inverse trigonometric functions arctan, arccot and their main properties, as well as several differentiation formulas of arctan and arccot.

- Bing Xie, Xiquan Liang, Fuguo Ge.
Inverse Trigonometric Functions
Arcsec and Arccosec,
Formalized Mathematics 16(2),
pages 159-165, 2008. MML Identifier:
SINCOS10

**Summary**: This article describes definitions of inverse trigonometric functions arcsec and arccosec, as well as their main properties.

- Noboru Endou, Keiko Narita, Yasunari Shidama.
The {L}ebesgue Monotone Convergence Theorem,
Formalized Mathematics 16(2),
pages 167-175, 2008. MML Identifier:
MESFUNC9

**Summary**: In this article we prove the Monotone Convergence Theorem \cite{Halmos74}.

- Grzegorz Bancerek.
Mizar Analysis of Algorithms: Algorithms over Integers,
Formalized Mathematics 16(2),
pages 177-194, 2008. MML Identifier:
AOFA_I00

**Summary**: This paper is a continuation of \cite{AOFA_000.ABS} and concerns if-while algebras over integers. In these algebras the only elementary instructions are assignment instructions. The instruction assigns to a (program) variable a value which is calculated for the current state according to some arithmetic expression. The expression may include variables, constants, and a limited number of arithmetic operations. States are functions from a given set of locations into integers. A variable is a function from the states into the locations and an expression is a function from the states into integers. Additional conditions (computability) limit the set of variables and expressions and, simultaneously, allow to write algorithms in a natural way (and to prove their correctness).\par As examples the proofs of full correctness of two Euclid algorithms (with modulo operation and subtraction) and algorithm of exponentiation by squaring are given.

- Yatsuka Nakamura, Kunio Oniumi, Wenpai Chang.
Invertibility of Matrices of Field Elements,
Formalized Mathematics 16(2),
pages 195-202, 2008. MML Identifier:
MATRIX14

**Summary**: In this paper the theory of invertibility of matrices of field elements (see e.g. \cite{Furuya:57}, \cite{Gantmacher:59}) is developed. The main purpose of this article is to prove that the left invertibility and the right invertibility are equivalent for a matrix of field elements. To prove this, we introduced a special transformation of matrix to some canonical forms. Other concepts as zero vector and base vectors of field elements are also introduced as a preparation.

- Marco Riccardi.
Ramsey's Theorem,
Formalized Mathematics 16(2),
pages 203-205, 2008. MML Identifier:
RAMSEY_1

**Summary**: The goal of this article is to formalize two versions of Ramsey's theorem. The theorems are not phrased in the usually pictorial representation of a coloured graph but use a set-theoretic terminology. After some useful lemma, the second section presents a generalization of Ramsey's theorem on infinite set closely following the book \cite{ThomasJech2002}. The last section includes the formalization of the theorem in a more known version (see \cite{PFTB}).

- Grzegorz Bancerek.
Towards the Construction
of a Model of {M}izar Concepts,
Formalized Mathematics 16(2),
pages 207-230, 2008. MML Identifier:
ABCMIZ_1

**Summary**: The aim of this paper is to develop a formal theory of Mizar linguistic concepts following the ideas from \cite{Bancerek:2003} and \cite{ABCMIZ_0.ABS}. The theory here presented is an abstract of the existing implementation of the Mizar system and is devoted to the formalization of Mizar expressions. The base idea behind the formalization is dependence on variables which is determined by variable-dependence (variables may depend on other variables). The dependence constitutes a Galois connection between opposite poset of dependence-closed set of variables and the sup-semilattice of widening of Mizar types (smooth type widening).\par In the paper the concepts strictly connected with Mizar expressions are formalized. Among them are quasi-loci, quasi-terms, quasi-adjectives, and quasi-types. The structural induction and operation of substitution are also introduced. The prefix {\it quasi} is used to indicate that some rules of construction of Mizar expressions may not be fulfilled. For example, variables, quasi-loci, and quasi-terms have no assigned types and, in result, there is no possibility to conduct type-checking of arguments. The other gaps concern inconsistent and out-of-context clusters of adjectives in types. Those rules are required in the Mizar {\it identification} process. However, the expression appearing in later processes of Mizar checker may not satisfy the rules. So, introduced apparatus is enough and adequate to describe data structures and algorithms from the Mizar checker (like {\it equational classes}).

- Jessica Enright, Piotr Rudnicki.
Helly Property for Subtrees,
Formalized Mathematics 16(2),
pages 91-96, 2008. MML Identifier:
HELLY
- Number 3
- Kazuhisa Ishida.
Model Checking. {P}art {II},
Formalized Mathematics 16(3),
pages 231-245, 2008. MML Identifier:
MODELC_2

**Summary**: This article provides the definition of linear temporal logic (LTL) and its properties relevant to model checking based on \cite{Clarke2000}. Mizar formalization of LTL language and satisfiability is based on \cite{ZF_LANG.ABS, ZF_MODEL.ABS}.

- Christoph Schwarzweller.
Modular Integer Arithmetic,
Formalized Mathematics 16(3),
pages 247-252, 2008. MML Identifier:
INT_6

**Summary**: In this article we show the correctness of integer arithmetic based on Chinese Remainder theorem as described e.g. in \cite{ModernComputerAlgebra}: Integers are transformed to finite sequences of modular integers, on which the arithmetic operations are performed. Retransformation of the results to the integers is then accomplished by means of the Chinese Remainder theorem. The method presented is a typical example for computing in homomorphic images.

- Tao Sun, Weibo Pan, Chenglong Wu, Xiquan Liang.
General Theory of Quasi-Commutative {BCI}-algebras,
Formalized Mathematics 16(3),
pages 253-258, 2008. MML Identifier:
BCIALG_5

**Summary**: It is known that commutative BCK-algebras form a variety, but BCK-algebras do not \cite{BCIAlgebras}. Therefore H.Yutani introduced the notion of quasi-commutative BCK-algebras. In this article we first present the notion and general theory of quasi-commutative BCI-algebras. Then we discuss the reduction of the type of quasi-commutative BCK-algebras and some special classes of quasi-commutative BCI-algebras.

- Karol P\k{a}k.
Block Diagonal Matrices,
Formalized Mathematics 16(3),
pages 259-267, 2008. MML Identifier:
MATRIXJ1

**Summary**: In this paper I present basic properties of block diagonal matrices over a set. In my approach the finite sequence of matrices in a block diagonal matrix is not restricted to square matrices. Moreover, the off-diagonal blocks need not be zero matrices, but also with another arbitrary fixed value.

- Karol P\k{a}k.
Linear Map of Matrices,
Formalized Mathematics 16(3),
pages 269-275, 2008. MML Identifier:
MATRLIN2

**Summary**: The paper is concerned with a generalization of concepts introduced in \cite{MATRLIN.ABS}, i.e. introduced are matrices of linear transformations over a finite-dimensional vector space. Introduced are linear transformations over a finite-dimensional vector space depending on a given matrix of the transformation. Finally, I prove that the rank of linear transformations over a finite-dimensional vector space is the same as the rank of the matrix of that transformation.

- El\.zbieta M\k{a}dra, Adam Grabowski.
Orthomodular Lattices,
Formalized Mathematics 16(3),
pages 277-282, 2008. MML Identifier:
ROBBINS4

**Summary**: The main result of the article is the solution to the problem of short axiomatizations of orthomodular ortholattices. Based on EQP/Otter results \cite{McCune:2005}, we gave a set of three equations which is equivalent to the classical, much longer equational basis of such a class. Also the basic example of the lattice which is not orthomodular, i.e. benzene (or $B_6$) is defined in two settings -- as a relational structure (poset) and as a lattice. \par As a preliminary work, we present the proofs of the dependence of other axiomatizations of ortholattices. The formalization of the properties of orthomodular lattices follows \cite{Beran:1984}.

- Yatsuka Nakamura, Hisashi Ito.
Basic Properties and Concept
of Selected Subsequence of Zero Based Finite Sequences,
Formalized Mathematics 16(3),
pages 283-288, 2008. MML Identifier:
AFINSQ_2

**Summary**: Here, we develop the theory of zero based finite sequences, which are sometimes, more useful in applications than normal one based finite sequences. The fundamental function Sgm is introduced as well as in case of normal finite sequences and other notions are also introduced. However, many theorems are a modification of old theorems of normal finite sequences, they are basically important and are necessary for applications. A new concept of selected subsequence is introduced. This concept came from the individual Ergodic theorem (see \cite{Halmos:1956}) and it is the preparation for its proof.

- Kazuhisa Ishida.
Model Checking. {P}art {II},
Formalized Mathematics 16(3),
pages 231-245, 2008. MML Identifier:
MODELC_2
- Number 4
- Karol P\k{a}k.
Eigenvalues of a Linear Transformation,
Formalized Mathematics 16(4),
pages 289-295, 2008. MML Identifier:
VECTSP11

**Summary**: The article presents well known facts about eigenvalues of linear transformation of a vector space (see \cite{HerstenWinter}). I formalize main dependencies between eigenvalues and the diagram of the matrix of a linear transformation over a finite-dimensional vector space. Finally, I formalize the subspace $\displaystyle\bigcup_{i=0}^\infty {\rm Ker}(f-\lambda I)^i$ called a generalized eigenspace for the eigenvalue $\lambda$ and show its basic properties.

- Karol P\k{a}k.
Jordan Matrix Decomposition,
Formalized Mathematics 16(4),
pages 297-303, 2008. MML Identifier:
MATRIXJ2

**Summary**: In this paper I present the Jordan Matrix Decomposition Theorem which states that an arbitrary square matrix $M$ over an algebraically closed field can be decomposed into the form $$M=SJS^{-1}$$ where $S$ is an invertible matrix and $J$ is a matrix in a Jordan canonical form, i.e. a special type of block diagonal matrix in which each block consists of Jordan blocks (see \cite{GolubWilkinson}).

- Noboru Endou, Keiko Narita, Yasunari Shidama.
Fatou's Lemma and the {L}ebesgue's Convergence Theorem,
Formalized Mathematics 16(4),
pages 305-309, 2008. MML Identifier:
MESFUN10

**Summary**: In this article we prove the Fatou's Lemma and Lebesgue's Convergence Theorem \cite{Halmos74}.

- Masahiko Yamazaki, Hiroshi Yamazaki, Yasunari Shidama.
Extended {R}iemann Integral of Functions
of Real Variable and One-sided {L}aplace Transform,
Formalized Mathematics 16(4),
pages 311-317, 2008. MML Identifier:
INTEGR10

**Summary**: In this article, we defined a variety of extended Riemann integrals and proved that such integration is linear. Furthermore, we defined the one-sided Laplace transform and proved the linearity of that operator.

- Keiko Narita, Noboru Endou, Yasunari Shidama.
Integral of Complex-Valued Measurable Function,
Formalized Mathematics 16(4),
pages 319-324, 2008. MML Identifier:
MESFUN6C

**Summary**: In this article, we formalized the notion of the integral of a complex-valued function considered as a sum of its real and imaginary parts. Then we defined the measurability and integrability in this context, and proved the linearity and several other basic properties of complex-valued measurable functions. The set of properties showed in this paper is based on \cite{MESFUNC6.ABS}, where the case of real-valued measurable functions is considered.

- Grzegorz Bancerek, Yasunari Shidama.
Introduction to Matroids,
Formalized Mathematics 16(4),
pages 325-332, 2008. MML Identifier:
MATROID0

**Summary**: The paper includes elements of the theory of matroids \cite{Welsh:1976}. The formalization is done according to \cite{Lipski}.

- Bing Xie, Xiquan Liang, Hongwei Li.
Partial Differentiation of Real Binary Functions,
Formalized Mathematics 16(4),
pages 333-338, 2008. MML Identifier:
PDIFF_2

**Summary**: In this article, we define two single-variable functions SVF1 and SVF2, then discuss partial differentiation of real binary functions by dint of one variable function SVF1 and SVF2. The main properties of partial differentiation are shown \cite{PDIFF_1.ABS}.

- Kazuhisa Ishida, Yasunari Shidama.
Model Checking. {P}art {III},
Formalized Mathematics 16(4),
pages 339-353, 2008. MML Identifier:
MODELC_3

**Summary**: This text includes verification of the basic algorithm in Simple On-the-fly Automatic Verification of Linear Temporal Logic (LTL). LTL formula can be transformed to Buchi automaton, and this transforming algorithm is mainly used at Simple On-the-fly Automatic Verification. In this article, we verified the transforming algorithm itself. At first, we prepared some definitions and operations for transforming. And then, we defined the Buchi automaton and verified the transforming algorithm.

- Xiaopeng Yue, Xiquan Liang.
Basic Properties of Circulant Matrices
and Anti-Circular Matrices,
Formalized Mathematics 16(4),
pages 355-360, 2008. MML Identifier:
MATRIX16

**Summary**: This article introduces definitions of circulant matrices, line- and column-circulant matrices as well as anti-circular matrices and describes their main properties.

- Yasushige Watase, Noboru Endou, Yasunari Shidama.
On ${L}^1$ Space Formed by
Real-Valued Partial Functions,
Formalized Mathematics 16(4),
pages 361-369, 2008. MML Identifier:
LPSPACE1

**Summary**: This article contains some definitions and properties refering to function spaces formed by partial functions defined over a measurable space. We formalized a function space, the so-called $L^1$ space and proved that the space turns out to be a normed space. The formalization of a real function space was given in \cite{FUNCSDOM.ABS}. The set of all function forms additive group. Here addition is defined by point-wise addition of two functions. However it is not true for partial functions. The set of partial functions does not form an additive group due to lack of right zeroed condition. Therefore, firstly we introduced a kind of a quasi-linear space, then, we introduced the definition of an equivalent relation of two functions which are almost everywhere equal ($=_{\rm a.e.}$), thirdly we formalized a linear space by taking the quotient of a quasi-linear space by the relation ($=_{\rm a.e.}$).

- Yuzhong Ding, Fuguo Ge, Chenglong Wu.
{BCI}-homomorphisms,
Formalized Mathematics 16(4),
pages 371-376, 2008. MML Identifier:
BCIALG_6

**Summary**: In this article the notion of the power of an element of BCI-algebra and its period in the book \cite{BCIAlgebras}, sections 1.4 to 1.5 are firstly given. Then the definition of BCI-homomorphism is defined and the fundamental theorem of homomorphism, the first isomorphism theorem and the second isomorphism theorem are proved following the book \cite{BCIAlgebras2}, section 1.6.

- Katsumi Wasaki.
Stability of the 4-2 Binary Addition
Circuit Cells. {P}art {I},
Formalized Mathematics 16(4),
pages 377-387, 2008. MML Identifier:
FTACELL1

**Summary**: To evaluate our formal verification method on a real-size calculation circuit, in this article, we continue to formalize the concept of the 4-2 Binary Addition Cell primitives (FTAs) to define the structures of calculation units for a very fast multiplication algorithm for VLSI implementation \cite{Vuillemin1983}. We define the circuit structure of four-types FTAs, TYPE-0 to TYPE-3, using the series constructions of the Generalized Full Adder Circuits (GFAs) that generalized adder to have for each positive and negative weights to inputs and outputs \cite{GFACIRC1.ABS}. We then successfully prove its circuit stability of the calculation outputs after four-steps. The motivation for this research is to establish a technique based on formalized mathematics and its applications for calculation circuits with high reliability.

- Fuguo Ge, Bing Xie.
Several Differentiation Formulas
of Special Functions. {P}art {VII},
Formalized Mathematics 16(4),
pages 389-399, 2008. MML Identifier:
FDIFF_11

**Summary**: In this article, we prove a series of differentiation identities \cite{Chemnitius:1956} involving the arctan and arccot functions and specific combinations of special functions including trigonometric and exponential functions.

- Hideki Sakurai, Hisayoshi Kunimune, Yasunari Shidama.
Open Mapping Theorem,
Formalized Mathematics 16(4),
pages 401-403, 2008. MML Identifier:
LOPBAN_6

**Summary**: In this article we formalize one of the most important theorems of linear operator theory the Open Mapping Theorem commonly used in a standard book such as \cite{miyadera:1972} in chapter 2.4.2. It states that a surjective continuous linear operator between Banach spaces is an open map.

- Karol P\k{a}k.
Eigenvalues of a Linear Transformation,
Formalized Mathematics 16(4),
pages 289-295, 2008. MML Identifier:
VECTSP11
**Volume 17, 2009**- Number 1
- Yatsuka Nakamura, Artur Korni{\l}owicz, Nagato Oya, Yasunari Shidama.
The Real Vector Spaces of
Finite Sequences are Finite Dimensional,
Formalized Mathematics 17(1),
pages 1-9, 2009. MML Identifier:
EUCLID_7

**Summary**: In this paper we show the finite dimensionality of real linear spaces with their carriers equal ${\cal R}^n$. We also give the standard basis of such spaces. For the set ${\cal R}^n$ we introduce the concepts of linear manifold subsets and orthogonal subsets. The cardinality of orthonormal basis of discussed spaces is proved to equal $n$.

- Bo Li, Yanping Zhuang, Bing Xie, Pan Wang.
Several Integrability Formulas
of Some Functions, Orthogonal Polynomials and Norm Functions,
Formalized Mathematics 17(1),
pages 11-21, 2009. MML Identifier:
INTEGRA9

**Summary**: In this article, we give several integrability formulas of some functions including the trigonometric function and the index function \cite{Chen:1978}. We also give the definitions of the orthogonal polynomial and norm function, and some of their important properties \cite{Renhong:1999}.

- Bo Li, Yanping Zhuang, Yanhong Men, Xiquan Liang.
Several Integrability
Formulas of Special Functions. {P}art {II},
Formalized Mathematics 17(1),
pages 23-35, 2009. MML Identifier:
INTEGR11

**Summary**: In this article, we give several differentiation and integrability formulas of special and composite functions including the trigonometric function, the hyperbolic function and the polynomial function \cite{Chen:1978}.

- Mitsuru Jitsukawa, Pauline N. Kawamoto, Yasunari Shidama, Yatsuka Nakamura.
Cell {P}etri Net Concepts,
Formalized Mathematics 17(1),
pages 37-42, 2009. MML Identifier:
PETRI_2

**Summary**: Based on the Petri net definitions and theorems already formalized in \cite{PETRI.ABS}, with this article, we developed the concept of ``Cell Petri Nets". It is based on \cite{Kawamoto-Nakamura:1996}. In a cell Petri net we introduce the notions of colors and colored states of a Petri net, connecting mappings for linking two Petri nets, firing rules for transitions, and the synthesis of two or more Petri nets.

- Artur Korni{\l}owicz.
Arithmetic Operations on Functions
from Sets into Functional Sets,
Formalized Mathematics 17(1),
pages 43-60, 2009. MML Identifier:
VALUED_2

**Summary**: In this paper we introduce sets containing number-valued functions. Different arithmetic operations on maps between any set and such functional sets are later defined.

- Yatsuka Nakamura, Artur Korni{\l}owicz, Nagato Oya, Yasunari Shidama.
The Real Vector Spaces of
Finite Sequences are Finite Dimensional,
Formalized Mathematics 17(1),
pages 1-9, 2009. MML Identifier:
EUCLID_7
- Number 2
- Bo Li, Pan Wang, Xiquan Liang, Yanping Zhuang.
Some Operations on Quaternion Numbers,
Formalized Mathematics 17(2),
pages 61-65, 2009. MML Identifier:
QUATERN3

**Summary**: In this article, we give some equality and basic theorems about quaternion numbers, and some special operations.

- Chanapat Pacharapokin, Hiroshi Yamazaki, Yasunari Shidama, Yatsuka Nakamura.
Complex Function Differentiability,
Formalized Mathematics 17(2),
pages 67-72, 2009. MML Identifier:
CFDIFF_1

**Summary**: For a complex valued function defined on its domain in complex numbers the differentiability in a single point and on a subset of the domain is presented. The main elements of differential calculus are developed. The algebraic properties of differential complex functions are shown.

- Agnes Doll.
Kolmogorov's Zero-One Law,
Formalized Mathematics 17(2),
pages 73-77, 2009. MML Identifier:
KOLMOG01

**Summary**: This article presents the proof of Kolmogorov's zero-one law in probability theory. The independence of a family of $\sigma$-fields is defined and basic theorems on it are given.

- Bing Xie, Xiquan Liang, Xiuzhuan Shen.
Second-Order Partial Differentiation of Real Binary Functions,
Formalized Mathematics 17(2),
pages 79-87, 2009. MML Identifier:
PDIFF_3

**Summary**: In this article we define second-order partial differentiation of real binary functions and discuss the relation of second-order partial derivatives and partial derivatives defined in \cite{PDIFF_2.ABS}.

- Keiko Narita, Noboru Endou, Yasunari Shidama.
The Measurability of Complex-Valued Functional Sequences,
Formalized Mathematics 17(2),
pages 89-97, 2009. MML Identifier:
MESFUN7C

**Summary**: In this article, we formalized the measurability of complex-valued functional sequences. First, we proved the measurability of the limits of real-valued functional sequences. Next, we defined complex-valued functional sequences dividing real part into imaginary part. Then using the former theorems, we proved the measurability of each part. Lastly, we proved the measurability of the limits of complex-valued functional sequences. We also showed several properties of complex-valued measurable functions. In addition, we proved properties of complex-valued simple functions.

- Artur Korni{\l}owicz.
Collective Operations on Number-Membered Sets,
Formalized Mathematics 17(2),
pages 99-115, 2009. MML Identifier:
MEMBER_1

**Summary**: The article starts with definitions of sets of opposite and inverse numbers of a given number membered set. Next, collective addition, subtraction, multiplication and division of two sets are defined. Complex numbers cases and extended real numbers ones are introduced separately and unified for reals. Shortcuts for singletons cases are also defined.

- Marco Riccardi.
Solution of Cubic and Quartic Equations,
Formalized Mathematics 17(2),
pages 117-122, 2009. MML Identifier:
POLYEQ_5

**Summary**: In this article, the principal $n$-th root of a complex number is defined, the Vieta's formulas for polynomial equations of degree 2, 3 and 4 are formalized. The solution of quadratic equations, the Cardan's solution of cubic equations and the Descartes-Euler solution of quartic equations in terms of their complex coefficients are also presented \cite{Korn}.

- Marco Riccardi.
The Perfect Number Theorem and {W}ilson's Theorem,
Formalized Mathematics 17(2),
pages 123-128, 2009. MML Identifier:
NAT_5

**Summary**: This article formalizes proofs of some elementary theorems of number theory (see \cite{PFTB, LeVeque}): Wilson's theorem (that $n$ is prime iff $n > 1$ and $(n-1)! \cong -1$ (mod $n$)), that all primes (1 mod 4) equal the sum of two squares, and two basic theorems of Euclid and Euler about perfect numbers. The article also formally defines Euler's sum of divisors function $\phi$, proves that $\phi$ is multiplicative and that $\sum_{k|n} \phi (k) = n$.

- Hiroyuki Okazaki, Yasunari Shidama.
Probability on Finite Set and Real-Valued Random Variables,
Formalized Mathematics 17(2),
pages 129-136, 2009. MML Identifier:
RANDOM_1

**Summary**: In the various branches of science, probability and randomness provide us with useful theoretical frameworks. The {\it Formalized Mathematics} has already published some articles concerning the probability: \cite{PROB_2.ABS}, \cite{PROB_1.ABS}, \cite{RPR_1.ABS}, and \cite{PROB_4.ABS}. In order to apply those articles, we shall give some theorems concerning the probability and the real-valued random variables to prepare for further studies.

- Keiko Narita, Noboru Endou, Yasunari Shidama.
Lebesgue's Convergence Theorem
of Complex-Valued Function,
Formalized Mathematics 17(2),
pages 137-145, 2009. MML Identifier:
MESFUN9C

**Summary**: In this article, we formalized Lebesgue's Convergence theorem of complex-valued function. We proved Lebesgue's Convergence Theorem of real-valued function using the theorem of extensional real-valued function. Then applying the former theorem to real part and imaginary part of complex-valued functional sequences, we proved Lebesgue's Convergence Theorem of complex-valued function. We also defined partial sums of real-valued functional sequences and complex-valued functional sequences and showed their properties. In addition, we proved properties of complex-valued simple functions.

- Hiroshi Yamazaki, Yasunari Shidama, Chanapat Pacharapokin, Yatsuka Nakamura.
The {C}auchy-{R}iemann Differential
Equations of Complex Functions,
Formalized Mathematics 17(2),
pages 147-149, 2009. MML Identifier:
CFDIFF_2

**Summary**: In this article we prove Cauchy-Riemann differential equations of complex functions. These theorems give necessary and sufficient condition for differentiable function.

- Kenichi Arai, Hiroyuki Okazaki.
Properties of Primes and Multiplicative Group of a Field,
Formalized Mathematics 17(2),
pages 151-155, 2009. MML Identifier:
GR_CY_3

**Summary**: In the \cite{INT_7.ABS} has been proven that the multiplicative group ${\mathbb Z}/p{{\mathbb Z}^\ast}$ is a cyclic group. Likewise, finite subgroup of the multiplicative group of a field is a cyclic group. However, finite subgroup of the multiplicative group of a field being a cyclic group has not yet been proven. Therefore, it is of importance to prove that finite subgroup of the multiplicative group of a field is a cyclic group. Meanwhile, in cryptographic system like RSA, in which security basis depends upon the difficulty of factorization of given numbers into prime factors, it is important to employ integers that are difficult to be factorized into prime factors. If both $p$ and $2p+1$ are prime numbers, we call $p$ as Sophie Germain prime, and $2p+1$ as safe prime. It is known that the product of two safe primes is a composite number that is difficult for some factoring algorithms to factorize into prime factors. In addition, safe primes are also important in cryptography system because of their use in discrete logarithm based techniques like Diffie-Hellman key exchange. If $p$ is a safe prime, the multiplicative group of numbers modulo $p$ has a subgroup of large prime order. However, no definitions have not been established yet with the safe prime and Sophie Germain prime. So it is important to give definitions of the Sophie Germain prime and safe prime. In this article, we prove finite subgroup of the multiplicative group of a field is a cyclic group, and, further, define the safe prime and Sophie Germain prime, and prove several facts about them. In addition, we define Mersenne number (${M}_{n}$), and some facts about Mersenne numbers and prime numbers are proven.

- Noboru Endou, Hiroyuki Okazaki, Yasunari Shidama.
Hopf Extension Theorem of Measure,
Formalized Mathematics 17(2),
pages 157-162, 2009. MML Identifier:
MEASURE8

**Summary**: The authors have presented some articles about Lebesgue type integration theory. In our previous articles \cite{MESFUNC9.ABS, MESFUNC5.ABS, RINFSUP2.ABS}, we assumed that some $\sigma$-additive measure existed and that a function was measurable on that measure. However the existence of such a measure is not trivial. In general, because the construction of a finite additive measure is comparatively easy, to induce a $\sigma$-additive measure a finite additive measure is used. This is known as an E.~Hopf's extension theorem of measure \cite{Halmos74}.

- Micha{\l} Trybulec.
Labelled State Transition Systems,
Formalized Mathematics 17(2),
pages 163-171, 2009. MML Identifier:
REWRITE3

**Summary**: This article introduces labelled state transition systems, where transitions may be labelled by words from a given alphabet. Reduction relations from \cite{REWRITE1.ABS} are used to define transitions between states, acceptance of words, and reachable states. Deterministic transition systems are also defined.

- Hiroyuki Okazaki.
Probability on Finite and Discrete Set
and Uniform Distribution,
Formalized Mathematics 17(2),
pages 173-178, 2009. MML Identifier:
DIST_1

**Summary**: A pseudorandom number generator plays an important role in practice in computer science. For example: computer simulations, cryptology, and so on. A pseudorandom number generator is an algorithm to generate a sequence of numbers that is indistinguishable from the true random number sequence. In this article, we shall formalize the ``Uniform Distribution" that is the idealized set of true random number sequences. The basic idea of our formalization is due to \cite{CINTA}.

- Keiichi Miyajima, Yasunari Shidama.
Riemann Integral of Functions
from $\mathbb{R}$ into $\calR^n$,
Formalized Mathematics 17(2),
pages 179-185, 2009. MML Identifier:
INTEGR15

**Summary**: In this article, we define the Riemann Integral of functions from $\mathbb{R}$ into $\calR^n$, and prove the linearity of this operator. The presented method is based on \cite{Murray:1974}.

- Bo Li, Yanhong Men.
Basic Properties of Even and Odd Functions,
Formalized Mathematics 17(2),
pages 187-192, 2009. MML Identifier:
FUNCT_8

**Summary**: In this article we present definitions, basic properties and some examples of even and odd functions \cite{Chen:1978}.

- Micha{\l} Trybulec.
Equivalence of Deterministic and Nondeterministic
Epsilon Automata,
Formalized Mathematics 17(2),
pages 193-199, 2009. MML Identifier:
FSM_3

**Summary**: Based on concepts introduced in \cite{REWRITE3.ABS}, semiautomata and left-languages, automata and right-languages, and langauges accepted by automata are defined. The powerset construction is defined for transition systems, semiautomata and automata. Finally, the equivalence of deterministic and nondeterministic epsilon automata is shown.

- Bo Li, Pan Wang, Xiquan Liang, Yanping Zhuang.
Some Operations on Quaternion Numbers,
Formalized Mathematics 17(2),
pages 61-65, 2009. MML Identifier:
QUATERN3
- Number 3
- Karol P\k{a}k.
Basic Properties of Metrizable Topological Spaces,
Formalized Mathematics 17(3),
pages 201-205, 2009. MML Identifier:
METRIZTS

**Summary**: We continue Mizar formalization of general topology according to the book \cite{ENGEL:1} by Engelking. In the article, we present the final theorem of Section 4.1. Namely, the paper includes the formalization of theorems on the correspondence between the cardinalities of the basis and of some open subcover, and a discreet (closed) subspaces, and the weight of that metrizable topological space. We also define Lindel\"of spaces and state the above theorem in this special case. We also introduce the concept of separation among two subsets (see \cite{ENGEL:BM51}).

- Karol P\k{a}k.
Small Inductive Dimension of Topological Spaces,
Formalized Mathematics 17(3),
pages 207-212, 2009. MML Identifier:
TOPDIM_1

**Summary**: We present the concept and basic properties of the Menger-Urysohn small inductive dimension of topological spaces according to the books \cite{DUDA:BM61}. Namely, the paper includes the formalization of main theorems from Sections 1.1 and 1.2.

- Xiquan Liang, Dailu Li.
On Rough Subgroup of a Group,
Formalized Mathematics 17(3),
pages 213-217, 2009. MML Identifier:
GROUP_11

**Summary**: This article describes a rough subgroup with respect to a normal subgroup of a group, and some properties of the lower and the upper approximations in a group.

- Karol P\k{a}k.
Small Inductive Dimension of Topological Spaces. {P}art {II},
Formalized Mathematics 17(3),
pages 219-222, 2009. MML Identifier:
TOPDIM_2

**Summary**: In this paper we present basic properties of $n$-dimensional topological spaces according to the book \cite{ENGEL:BM51}. In the article the formalization of Section 1.5 is completed.

- Karol P\k{a}k.
Basic Properties of Metrizable Topological Spaces,
Formalized Mathematics 17(3),
pages 201-205, 2009. MML Identifier:
METRIZTS
- Number 4
- Piotr Rudnicki.
Dilworth's Decomposition Theorem for Posets,
Formalized Mathematics 17(4),
pages 223-232, 2009. MML Identifier:
DILWORTH

**Summary**: The following theorem is due to Dilworth \cite{Dilworth50}: Let $P$ be a partially ordered set. If the maximal number of elements in an independent subset (anti-chain) of $P$ is $k$, then $P$ is the union of $k$ chains (cliques).\par In this article we formalize an elegant proof of the above theorem for finite posets by Perles \cite{Perles63}. The result is then used in proving the case of infinite posets following the original proof of Dilworth \cite{Dilworth50}.\par A dual of Dilworth's theorem also holds: a poset with maximum clique $m$ is a union of $m$ independent sets. The proof of this dual fact is considerably easier; we follow the proof by Mirsky \cite{Mirsky71}. Mirsky states also a corollary that a poset of $r \times s + 1$ elements possesses a clique of size $r+1$ or an independent set of size $s+1$, or both. This corollary is then used to prove the result of Erd\H{o}s and Szekeres \cite{ES35}.\par Instead of using posets, we drop reflexivity and state the facts about antisymmetric and transitive relations.

- Masahiko Yamazaki, Hiroshi Yamazaki, Katsumi Wasaki, Yasunari Shidama.
Complex Integral,
Formalized Mathematics 17(4),
pages 233-236, 2009. MML Identifier:
INTEGR1C

**Summary**: In this article, we defined complex curve and complex integral. Then we have proved the linearity for the complex integral. Furthermore, we have proved complex integral of complex curve's connection is the sum of each complex integral of individual complex curve.

- Adam Grabowski, Magdalena Jastrz\k{e}bska.
On the Lattice of Intervals and Rough Sets,
Formalized Mathematics 17(4),
pages 237-244, 2009. MML Identifier:
INTERVA1

**Summary**: Rough sets, developed by Pawlak \cite{Pawlak1982}, are an important tool to describe a situation of incomplete or partially unknown information. One of the algebraic models deals with the pair of the upper and the lower approximation. Although usually the tolerance or the equivalence relation is taken into account when considering a rough set, here we rather concentrate on the model with the pair of two definable sets, hence we are close to the notion of an interval set. In this article, the lattices of rough sets and intervals are formalized. This paper, being essentially the continuation of \cite{ROUGHS_1.ABS}, is also a step towards the formalization of the algebraic theory of rough sets, as in \cite{Mousavi:2001} or \cite{Yao:1993}.

- Bo Li, Yanhong Men, Dailu Li, Xiquan Liang.
Basic Properties of Periodic Functions,
Formalized Mathematics 17(4),
pages 245-248, 2009. MML Identifier:
FUNCT_9

**Summary**: In this article we present definitions, basic properties and some examples of periodic functions according to \cite{Chen:1978}.

- Grzegorz Bancerek.
Epsilon Numbers and {C}antor Normal Form,
Formalized Mathematics 17(4),
pages 249-256, 2009. MML Identifier:
ORDINAL5

**Summary**: An epsilon number is a transfinite number which is a fixed point of an exponential map: $\omega^\varepsilon = \varepsilon$. The formalization of the concept is done with use of the tetration of ordinals (Knuth's arrow notation, $\uparrow\uparrow$). Namely, the ordinal indexing of epsilon numbers is defined as follows: $$\varepsilon_0 = \omega\mathop{\uparrow\uparrow}\omega,\ \ \ \varepsilon_{\alpha+1} = \varepsilon_\alpha\mathop{\uparrow\uparrow}\omega,$ and for limit ordinal $\lambda$: $$\varepsilon_\lambda = \lim_{\alpha<\lambda}\varepsilon_\alpha = \bigcup_{\alpha<\lambda}\varepsilon_\alpha.$$ Tetration stabilizes at $\omega$: $$\alpha\mathop{\uparrow\uparrow}\beta = \alpha\mathop{\uparrow\uparrow}\omega \ \ \ {\rm for}\ \ \ \alpha\neq 0\ \ \ {\rm and}\ \ \ \beta\geq\omega.$$ Every ordinal number $\alpha$ can be uniquely written as $$n_1\omega^{\beta_1} + n_2\omega^{\beta_2} + \cdots + n_k\omega^{\beta_k},$$ where $k$ is a natural number, $n_1$, $n_2$, $\ldots$, $n_k$ are positive integers, and $\beta_1 > \beta_2 > \ldots > \beta_k$ are ordinal numbers ($\beta_k = 0$). This decomposition of $\alpha$ is called the Cantor Normal Form of $\alpha$.

- Piotr Rudnicki.
Dilworth's Decomposition Theorem for Posets,
Formalized Mathematics 17(4),
pages 223-232, 2009. MML Identifier:
DILWORTH
**Volume 18, 2010**- Number 1
- Xiquan Liang, Piqing Zhao, Ou Bai.
Vector Functions and their
Differentiation Formulas in 3-dimensional {E}uclidean Spaces,
Formalized Mathematics 18(1),
pages 1-10, 2010. MML Identifier:
EUCLID_8

**Summary**: In this article, we first extend several basic theorems of the operation of vector in 3-dimensional Euclidean spaces. Then three unit vectors: $e1$, $e2$, $e3$ and the definition of vector function in the same spaces are introduced. By dint of unit vector the main operation properties as well as the differentiation formulas of vector function are shown \cite{Spiegel:1959}.

- Katuhiko Kanazashi, Noboru Endou, Yasunari Shidama.
Banach Algebra of Continuous
Functionals and the Space of Real-Valued Continuous Functionals with Bounded
Support,
Formalized Mathematics 18(1),
pages 11-16, 2010. MML Identifier:
C0SP2

**Summary**: In this article, we give a definition of a functional space which is constructed from all continuous functions defined on a compact topological space. We prove that this functional space is a Banach algebra. Next, we give a definition of a function space which is constructed from all real-valued continuous functions with bounded support. We prove that this function space is a real normed space.

- Marco Riccardi.
Free Magmas,
Formalized Mathematics 18(1),
pages 17-26, 2010. MML Identifier:
ALGSTR_4

**Summary**: This article introduces the free magma $M(X)$ constructed on a set $X$ \cite{BourbakiAlgI}. Then, we formalize some theorems about $M(X)$: if $f$ is a function from the set $X$ to a magma $N$, the free magma $M(X)$ has a unique extension of $f$ to a morphism of $M(X)$ into $N$ and every magma is isomorphic to a magma generated by a set $X$ under a set of relators on $M(X)$. In doing it, the article defines the stable subset under the law of composition of a magma, the submagma, the equivalence relation compatible with the law of composition and the equivalence kernel of a function. We also introduce some schemes on the recursive function.

- Bo Li, Na Ma.
Integrability Formulas. {P}art {I},
Formalized Mathematics 18(1),
pages 27-37, 2010. MML Identifier:
INTEGR12

**Summary**: In this article, we give several differentiation and integrability formulas of special and composite functions including the trigonometric function, and the polynomial function.

- Takao Inou\'e, Bing Xie, Xiquan Liang.
Partial Differentiation of Real Ternary Functions,
Formalized Mathematics 18(1),
pages 39-46, 2010. MML Identifier:
PDIFF_4

**Summary**: In this article, we shall extend the result of \cite{PDIFF_2.ABS} to discuss partial differentiation of real ternary functions (refer to \cite{PDIFF_1.ABS} and \cite{Rudin:1976} for partial differentiation).

- Kazuhisa Ishida, Yasunari Shidama.
Fixpoint Theorem for Continuous Functions on Chain-Complete Posets,
Formalized Mathematics 18(1),
pages 47-51, 2010. MML Identifier:
POSET_1

**Summary**: This text includes the definition of chain-complete poset, fix-point theorem on it, and the definition of the function space of continuous functions on chain-complete posets \cite{Winskel:1993}.

- Dailu Li, Xiquan Liang, Yanhong Men.
Nilpotent Groups,
Formalized Mathematics 18(1),
pages 53-56, 2010. MML Identifier:
GRNILP_1

**Summary**: This article describes the concept of the nilpotent group and some properties of the nilpotent groups.

- Xiquan Liang, Ling Tang.
Difference and Difference Quotient. {P}art {III},
Formalized Mathematics 18(1),
pages 57-64, 2010. MML Identifier:
DIFF_3

**Summary**: In this article, we give some important theorems of forward difference, backward difference, central difference and difference quotient and forward difference, backward difference, central difference and difference quotient formulas of some special functions.

- Grzegorz Bancerek.
A Model of {M}izar Concepts -- Unification,
Formalized Mathematics 18(1),
pages 65-75, 2010. MML Identifier:
ABCMIZ_A

**Summary**: The aim of this paper is to develop a formal theory of Mizar linguistic concepts following the ideas from \cite{Bancerek:2003} and \cite{ABCMIZ_1.ABS}. The theory presented is an abstraction from the existing implementation of the Mizar system and is devoted to the formalization of Mizar expressions. The concepts formalized here are: standarized constructor signature, arity-rich signatures, and the unification of Mizar expressions.

- Magdalena Jastrz\k{e}bska.
Representation of the {F}ibonacci
and {L}ucas Numbers in Terms of Floor and Ceiling,
Formalized Mathematics 18(1),
pages 77-80, 2010. MML Identifier:
FIB_NUM4

**Summary**: In the paper we show how to express the Fibonacci numbers and Lucas numbers using the floor and ceiling operations.

- Artur Korni{\l}owicz.
The Correspondence Between $n$-dimensional
{E}uclidean Space and the Product of $n$ Real Lines,
Formalized Mathematics 18(1),
pages 81-85, 2010. MML Identifier:
EUCLID_9

**Summary**: In the article we prove that a family of open $n$-hypercubes is a basis of $n$-dimensional Euclidean space. The equality of the space and the product of $n$ real lines has been proven.

- Karol P\k{a}k.
Affine Independence in Vector Spaces,
Formalized Mathematics 18(1),
pages 87-93, 2010. MML Identifier:
RLAFFIN1

**Summary**: In this article we describe the notion of affinely independent subset of a real linear space. First we prove selected theorems concerning operations on linear combinations. Then we introduce affine independence and prove the equivalence of various definitions of this notion. We also introduce the notion of the affine hull, i.e. a subset generated by a set of vectors which is an intersection of all affine sets including the given set. Finally, we introduce and prove selected properties of the barycentric coordinates.

- Karol P\k{a}k.
Abstract Simplicial Complexes,
Formalized Mathematics 18(1),
pages 95-106, 2010. MML Identifier:
SIMPLEX0

**Summary**: In this article we define the notion of abstract simplicial complexes and operations on them. We introduce the following basic notions: simplex, face, vertex, degree, skeleton, subdivision and substructure, and prove some of their properties.

- Xiquan Liang, Piqing Zhao, Ou Bai.
Vector Functions and their
Differentiation Formulas in 3-dimensional {E}uclidean Spaces,
Formalized Mathematics 18(1),
pages 1-10, 2010. MML Identifier:
EUCLID_8
- Number 2
- Keiichi Miyajima, Takahiro Kato.
The Sum and Product of Finite
Sequences of Complex Numbers,
Formalized Mathematics 18(2),
pages 107-111, 2010. MML Identifier:
RVSUM_2

**Summary**: This article extends the \cite{RVSUM_1.ABS}. We define the sum and the product of the sequence of complex numbers, and formalize these theorems. Our method refers to the \cite{Keith:1984}.

- Takao Inou\'e.
Second-Order
Partial Differentiation of Real Ternary Functions,
Formalized Mathematics 18(2),
pages 113-127, 2010. MML Identifier:
PDIFF_5

**Summary**: In this article, we shall extend the result of \cite{PDIFF_3.ABS} to discuss second-order partial differentiation of real ternary functions (refer to \cite{PDIFF_1.ABS} and \cite{Rudin:1976} for partial differentiation).

- Bo Li, Na Ma, Xiquan Liang.
Integrability Formulas. {P}art {II},
Formalized Mathematics 18(2),
pages 129-141, 2010. MML Identifier:
INTEGR13

**Summary**: In this article, we give several differentiation and integrability formulas of special and composite functions including trigonometric function, and polynomial function.

- Bo Li, Na Ma.
Integrability Formulas. {P}art {III},
Formalized Mathematics 18(2),
pages 143-157, 2010. MML Identifier:
INTEGR14

**Summary**: In this article, we give several differentiation and integrability formulas of composite trigonometric function.

- Keiichi Miyajima, Takahiro Kato.
The Sum and Product of Finite
Sequences of Complex Numbers,
Formalized Mathematics 18(2),
pages 107-111, 2010. MML Identifier:
RVSUM_2
- Number 3
- Yasushige Watase, Noboru Endou, Yasunari Shidama.
On ${L}^p$ Space Formed by Real-Valued Partial Functions,
Formalized Mathematics 18(3),
pages 159-169, 2010. MML Identifier:
LPSPACE2

**Summary**: This article is the continuation of \cite{LPSPACE1.ABS}. We define the set of $L^p$ integrable functions -- the set of all partial functions whose absolute value raised to the $p$-th power is integrable. We show that $L^p$ integrable functions form the $L^p$ space. We also prove Minkowski's inequality, H\"{o}lder's inequality and that $L^p$ space is Banach space (\cite{Halmos74}, \cite{Rudin:2}).

- Artur Korni{\l}owicz.
Miscellaneous Facts about Open Functions and Continuous Functions,
Formalized Mathematics 18(3),
pages 171-174, 2010. MML Identifier:
TOPS_4

**Summary**: In this article we give definitions of open functions and continuous functions formulated in terms of ``balls" of given topological spaces.

- Artur Korni{\l}owicz.
On the Continuity of Some Functions,
Formalized Mathematics 18(3),
pages 175-183, 2010. MML Identifier:
TOPREALC

**Summary**: We prove that basic arithmetic operations preserve continuity of functions.

- Karol P\k{a}k.
The Geometric Interior in Real Linear Spaces,
Formalized Mathematics 18(3),
pages 185-188, 2010. MML Identifier:
RLAFFIN2

**Summary**: We introduce the notions of the geometric interior and the centre of mass for subsets of real linear spaces. We prove a number of theorems concerning these notions which are used in the theory of abstract simplicial complexes.

- Yasushige Watase, Noboru Endou, Yasunari Shidama.
On ${L}^p$ Space Formed by Real-Valued Partial Functions,
Formalized Mathematics 18(3),
pages 159-169, 2010. MML Identifier:
LPSPACE2
- Number 4
- Karol P\k{a}k.
Sperner's Lemma,
Formalized Mathematics 18(4),
pages 189-196, 2010. MML Identifier:
SIMPLEX1

**Summary**: In this article we introduce and prove properties of simplicial complexes in real linear spaces which are necessary to formulate Sperner's lemma. The lemma states that for a function $f$, which for an arbitrary vertex $v$ of the barycentric subdivision $\mathcal{B}$ of simplex $\mathcal{K}$ assigns some vertex from a face of $\mathcal{K}$ which contains $v$, we can find a simplex $S$ of $\mathcal{B}$ which satisfies $f(S)=\mathcal{K}$ (see \cite{DUDA:BM61}).

- Cezary Kaliszyk.
Counting Derangements, Non Bijective Functions and the Birthday Problem,
Formalized Mathematics 18(4),
pages 197-200, 2010. MML Identifier:
CARDFIN2

**Summary**: The article provides counting derangements of finite sets and counting non bijective functions. We provide a recursive formula for the number of derangements of a finite set, together with an explicit formula involving the number $e$. We count the number of non-one-to-one functions between to finite sets and perform a computation to give explicitely a formalization of the birthday problem. The article is an extension of \cite{CARD_FIN.ABS}.

- Keiichi Miyajima, Takahiro Kato, Yasunari Shidama.
Riemann Integral of Functions $\mathbb{R}$ into $\mathbb{C}$,
Formalized Mathematics 18(4),
pages 201-206, 2010. MML Identifier:
INTEGR16

**Summary**: In this article, we define the Riemann Integral on functions $\mathbb{R}$ into $\mathbb{C}$ and proof the linearity of this operator. Especially, the Riemann integral of complex functions is constituted by the redefinition about the Riemann sum of complex numbers. Our method refers to the \cite{Murray:1974}.

- Takao Inou\'e, Noboru Endou, Yasunari Shidama.
Differentiation of Vector-Valued
Functions on $n$-Dimensional Real Normed Linear Spaces,
Formalized Mathematics 18(4),
pages 207-212, 2010. MML Identifier:
PDIFF_6

**Summary**: In this article, we define and develop differentiation of vector-valued functions on $n$-dimensional real normed linear spaces (refer to \cite{Rudin:1976} and \cite{Schwartz:1981}).

- Hiroyuki Okazaki, Yasunari Shidama.
{P}robability Measure on Discrete Spaces
and Algebra of Real-Valued Random Variables,
Formalized Mathematics 18(4),
pages 213-217, 2010. MML Identifier:
RANDOM_2

**Summary**: In this article we continue formalizing probability and randomness started in \cite{RANDOM_1.ABS}, where we formalized some theorems concerning the probability and real-valued random variables. In this paper we formalize the variance of a random variable and prove Chebyshev's inequality. Next we formalize the product probability measure on the Cartesian product of discrete spaces. In the final part of this article we define the algebra of real-valued random variables.

- Karol P\k{a}k.
Sperner's Lemma,
Formalized Mathematics 18(4),
pages 189-196, 2010. MML Identifier:
SIMPLEX1
**Volume 19, 2011**- Number 1
- Takao Inou\'e, Adam Naumowicz, Noboru Endou, Yasunari Shidama.
Partial Differentiation of Vector-Valued Functions
on $n$-Dimensional Real Normed Linear Spaces,
Formalized Mathematics 19(1),
pages 1-9, 2011. MML Identifier:
PDIFF_7

**Summary**: In this article, we define and develop partial differentiation of vector-valued functions on $n$-dimensional real normed linear spaces (refer to \cite{Rudin:1976} and \cite{Schwartz:1981}).

- Xiquan Liang, Dailu Li.
Some Properties of $p$-Groups and Commutative $p$-Groups,
Formalized Mathematics 19(1),
pages 11-15, 2011. MML Identifier:
GROUPP_1

**Summary**: This article describes some properties of $p$-groups and some properties of commutative $p$-groups.

- Keiichi Miyajima, Takahiro Kato, Yasunari Shidama.
Riemann Integral of Functions from $\mathbb{R}$ into Real Normed Space,
Formalized Mathematics 19(1),
pages 17-22, 2011. MML Identifier:
INTEGR18

**Summary**: In this article, we define the Riemann integral on functions from $\mathbb R$ into real normed space and prove the linearity of this operator. As a result, the Riemann integration can be applied to a wider range of functions. The proof method follows the \cite{Murray:1974}.

- Hiroyuki Okazaki, Kenichi Arai, Yasunari Shidama.
Normal Subgroup of Product of Groups,
Formalized Mathematics 19(1),
pages 23-26, 2011. MML Identifier:
GROUP_12

**Summary**: In \cite{GROUP_7.ABS} it was formalized that the direct product of a family of groups gives a new group. In this article, we formalize that for all $j \in I$, the group $G =\Pi_{i \in I} G_i$ has a normal subgroup isomorphic to $G_j$. Moreover, we show some relations between a family of groups and its direct product.

- Piotr Rudnicki, Lorna Stewart.
{T}he {M}ycielskian of a Graph,
Formalized Mathematics 19(1),
pages 27-34, 2011. MML Identifier:
MYCIELSK

**Summary**: Let $\omega(G)$ and $\chi(G)$ be the clique number and the chromatic number of a graph $G$. Mycielski~\cite{Mycielski55} presented a construction that for any $n$ creates a graph $M_n$ which is triangle-free ($\omega(G) = 2$) with $\chi(G) > n$. The starting point is the complete graph of two vertices ($K_2$). $M_{(n+1)}$ is obtained from $M_n$ through the operation $\mu(G)$ called the Mycielskian of a graph $G$. \par We first define the operation $\mu(G)$ and then show that $\omega(\mu(G)) = \omega(G)$ and $\chi(\mu(G)) = \chi(G)+1$. This is done for arbitrary graph $G$, see also~\cite{LUP95}. Then we define the sequence of graphs $M_n$ each of exponential size in $n$ and give their clique and chromatic numbers.

- Xiquan Liang, Ling Tang, Xichun Jiang.
{D}ifference and Difference Quotient. {P}art {IV},
Formalized Mathematics 19(1),
pages 35-39, 2011. MML Identifier:
DIFF_4

**Summary**: In this article, we give some important theorems of forward difference, backward difference, central difference and difference quotient and forward difference, backward difference, central difference and difference quotient formulas of some special functions.

- Marco Riccardi.
The Definition of Topological Manifolds,
Formalized Mathematics 19(1),
pages 41-44, 2011. MML Identifier:
MFOLD_1

**Summary**: This article introduces the definition of $n$-locally Euclidean topological spaces and topological manifolds \cite{JLee2000}.

- Hiroyuki Okazaki, Noboru Endou, Yasunari Shidama.
More on Continuous Functions on Normed Linear Spaces,
Formalized Mathematics 19(1),
pages 45-49, 2011. MML Identifier:
NFCONT_3

**Summary**: In this article we formalize the definition and some facts about continuous functions from $\Bbb R$ into normed linear spaces \cite{Schwartz:1967}.

- Hiroyuki Okazaki, Noboru Endou, Yasunari Shidama.
Cartesian Products of Family of Real Linear Spaces,
Formalized Mathematics 19(1),
pages 51-59, 2011. MML Identifier:
PRVECT_3

**Summary**: In this article we introduced the isomorphism mapping between cartesian products of family of linear spaces \cite{BOURBAKI:1-5}. Those products had been formalized by two different ways, i.e., the way using the functor [:X,Y:] and ones using the functor ``product". By the same way, the isomorphism mapping was defined between Cartesian products of family of linear normed spaces also.

- Yuichi Futa, Hiroyuki Okazaki, Yasunari Shidama.
Formalization of Integral Linear Space,
Formalized Mathematics 19(1),
pages 61-64, 2011. MML Identifier:
RLVECT_X

**Summary**: In this article, we formalize integral linear spaces, that is a linear space with integer coefficients. Integral linear spaces are necessary for lattice problems, LLL (Lenstra-Lenstra-Lov\'{a}sz) base reduction algorithm that outputs short lattice base and cryptographic systems with lattice~\cite{Micci:2002}.

- Takao Inou\'e, Adam Naumowicz, Noboru Endou, Yasunari Shidama.
Partial Differentiation of Vector-Valued Functions
on $n$-Dimensional Real Normed Linear Spaces,
Formalized Mathematics 19(1),
pages 1-9, 2011. MML Identifier:
PDIFF_7
- Number 2
- Takao Inou\'e, Adam Naumowicz, Noboru Endou, Yasunari Shidama.
Partial Differentiation, Differentiation and Continuity
on $n$-Dimensional Real Normed Linear Spaces,
Formalized Mathematics 19(2),
pages 65-68, 2011. MML Identifier:
PDIFF_8

**Summary**: In this article, we aim to prove the characterization of differentiation by means of partial differentiation for vector-valued functions on $n$-dimensional real normed linear spaces (refer to \cite{Rudin:1976} and \cite{Schwartz:1981}).

- Hiroyuki Okazaki, Noboru Endou, Keiko Narita, Yasunari Shidama.
Differentiable Functions into Real Normed Spaces,
Formalized Mathematics 19(2),
pages 69-72, 2011. MML Identifier:
NDIFF_3

**Summary**: In this article, we formalize the differentiability of functions from the set of real numbers into a normed vector space \cite{CA}.

- Robin Nittka.
Conway's Games and Some of their Basic Properties,
Formalized Mathematics 19(2),
pages 73-81, 2011. MML Identifier:
CGAMES_1

**Summary**: We formulate a few basic concepts of J.\ H.\ Conway's theory of games based on his book~\cite{Conway:2001}. This is a first step towards formalizing Conway's theory of numbers into Mizar, which is an approach to proving the existence of a FIELD (i.e., a proper class that satisfies the axioms of a real-closed field) that includes the reals and ordinals, thus providing a uniform, independent and simple approach to these two constructions that does not go via the rational numbers and hence does for example not need the notion of a quotient field.\par In this first article on Conway's games, we provide a definition of games, their birthdays (or ranks), their trees (a notion which is not in Conway's book, but is useful as a tool), their negates and their signs, together with some elementary properties of these notions. If one is interested only in Conway's numbers, it would have been easier to define them directly, but going via the notion of a game is a more general approach in the sense that a number is a special instance of a game and that there is a rich theory of games that are not numbers.\par The main obstacle in formulating these topics in Mizar is that all definitions are highly recursive, which is not entirely simple to translate into the Mizar language. For example, according to Conway's definition, a game is an object consisting of left and right options which are themselves games, and this is by definition the only way to construct a game. This cannot directly be translated into Mizar, but a theorem is included in the article which proves that our definition is equivalent to Conway's.

- Grzegorz Bancerek.
Veblen Hierarchy,
Formalized Mathematics 19(2),
pages 83-92, 2011. MML Identifier:
ORDINAL6

**Summary**: The Veblen hierarchy is an extension of the construction of epsilon numbers (fixpoints of the exponential map: $\omega^\epsilon = \epsilon$). It is a collection $\varphi_\alpha$ of the Veblen Functions where $\varphi_0(\beta) = \omega^\beta$ and $\varphi_1(\beta) = \epsilon_\beta$. The sequence of fixpoints of $\varphi_1$ function form $\varphi_2$, etc. For a limit non empty ordinal $\lambda$ the function $\varphi_\lambda$ is the sequence of common fixpoints of all functions $\varphi_\alpha$ where $\alpha < \lambda$.\par The Mizar formalization of the concept cannot be done directly as the Veblen functions are classes (not (small) sets). It is done with use of universal sets (Tarski classes). Namely, we define the Veblen functions in a given universal set and $\varphi_\alpha(\beta)$ as a value of Veblen function from the smallest universal set including $\alpha$ and $\beta$.

- Grzegorz Bancerek.
Sorting by Exchanging,
Formalized Mathematics 19(2),
pages 93-102, 2011. MML Identifier:
EXCHSORT

**Summary**: We show that exchanging of pairs in an array which are in incorrect order leads to sorted array. It justifies correctness of Bubble Sort, Insertion Sort, and Quicksort.

- Karol P\k{a}k.
{L}inear Transformations of {E}uclidean Topological Spaces,
Formalized Mathematics 19(2),
pages 103-108, 2011. MML Identifier:
MATRTOP1

**Summary**: We introduce linear transformations of Euclidean topological spaces given by a transformation matrix. Next, we prove selected properties and basic arithmetic operations on these linear transformations. Finally, we show that a linear transformation given by an invertible matrix is a homeomorphism.

- Karol P\k{a}k.
{L}inear Transformations of {E}uclidean Topological
Spaces. {P}art {II},
Formalized Mathematics 19(2),
pages 109-112, 2011. MML Identifier:
MATRTOP2

**Summary**: We prove a number of theorems concerning various notions used in the theory of continuity of barycentric coordinates.

- Mariusz Giero.
The Axiomatization of Propositional Linear Time Temporal Logic,
Formalized Mathematics 19(2),
pages 113-119, 2011. MML Identifier:
LTLAXIO1

**Summary**: The article introduces propositional linear time temporal logic as a formal system. Axioms and rules of derivation are defined. Soundness Theorem and Deduction Theorem are proved \cite{KroMer}.

- Katuhiko Kanazashi, Hiroyuki Okazaki, Yasunari Shidama.
Banach Algebra of Bounded Complex-Valued Functionals,
Formalized Mathematics 19(2),
pages 121-126, 2011. MML Identifier:
CC0SP1

**Summary**: In this article, we describe some basic properties of the Banach algebra which is constructed from all bounded complex-valued functionals.

- Takao Inou\'e, Adam Naumowicz, Noboru Endou, Yasunari Shidama.
Partial Differentiation, Differentiation and Continuity
on $n$-Dimensional Real Normed Linear Spaces,
Formalized Mathematics 19(2),
pages 65-68, 2011. MML Identifier:
PDIFF_8
- Number 3
- Artur Korni{\l}owicz.
{M}azur-{U}lam Theorem,
Formalized Mathematics 19(3),
pages 127-130, 2011. MML Identifier:
MAZURULM

**Summary**: The Mazur-Ulam theorem \cite{MazurUlam} has been formulated as two registrations: {\verb!cluster bijective isometric -> midpoints-preserving Function!} {\verb!of E,F;!} and {\verb!cluster isometric midpoints-preserving -> Affine Function!} {\verb!of E,F;!} A proof given by Jussi V\"{a}is\"{a}l\"{a} \cite{Jussi} has been formalized.

- Yuichi Futa, Hiroyuki Okazaki, Yasunari Shidama.
Set of Points on Elliptic Curve in Projective Coordinates,
Formalized Mathematics 19(3),
pages 131-138, 2011. MML Identifier:
EC_PF_1

**Summary**: In this article, we formalize a set of points on an elliptic curve over $\bf{GF}(p)$. Elliptic curve cryptography \cite{BSS99}, whose security is based on a difficulty of discrete logarithm problem of elliptic curves, is important for information security.

- Karol P\k{a}k.
Continuity of Barycentric Coordinates in {E}uclidean Topological Spaces,
Formalized Mathematics 19(3),
pages 139-144, 2011. MML Identifier:
RLAFFIN3

**Summary**: In this paper we present selected properties of barycentric coordinates in the Euclidean topological space. We prove the topological correspondence between a subset of an affine closed space of ${\cal E}^n$ and the set of vectors created from barycentric coordinates of points of this subset.

- Karol P\k{a}k.
{B}rouwer Fixed Point Theorem for Simplexes,
Formalized Mathematics 19(3),
pages 145-150, 2011. MML Identifier:
SIMPLEX2

**Summary**: In this article we prove the Brouwer fixed point theorem for an arbitrary simplex which is the convex hull of its $n + 1$ affinely indepedent vertices of ${\cal E}^n$. First we introduce the Lebesgue number, which for an arbitrary open cover of a compact metric space $\mathfrak{M}$ is a positive real number so that any ball of about such radius must be completely contained in a member of the cover. Then we introduce the notion of a bounded simplicial complex and the diameter of a bounded simplicial complex. We also prove the estimation of diameter decrease which is connected with the barycentric subdivision. Finally, we prove the Brouwer fixed point theorem and compute the small inductive dimension of ${\cal E}^n$. This article is based on \cite{DUDA:BM61}.

- Karol P\k{a}k.
Brouwer Fixed Point Theorem in the General Case,
Formalized Mathematics 19(3),
pages 151-153, 2011. MML Identifier:
BROUWER2

**Summary**: In this article we prove the Brouwer fixed point theorem for an arbitrary convex compact subset of ${\cal E}^n$ with a non empty interior. This article is based on \cite{SIEKLUCKI:BM53}.

- Marco B. Caminati.
Preliminaries to Classical First Order Model Theory,
Formalized Mathematics 19(3),
pages 155-167, 2011. MML Identifier:
FOMODEL0

**Summary**: First of a series of articles laying down the bases for classical first order model theory. These articles introduce a framework for treating arbitrary languages with equality. This framework is kept as generic and modular as possible: both the language and the derivation rule are introduced as a type, rather than a fixed functor; definitions and results regarding syntax, semantics, interpretations and sequent derivation rules, respectively, are confined to separate articles, to mark out the hierarchy of dependences among different definitions and constructions. \par As an application limited to countable languages, satisfiability theorem and a full version of the G\"odel completeness theorem are delivered, with respect to a fixed, remarkably thrifty, set of correct rules. Besides the self-referential significance for the Mizar project itself of those theorems being formalized with respect to a generic, equality-furnished, countable language, this is the first step to work out other milestones of model theory, such as Lowenheim-Skolem and compactness theorems. Being the receptacle of all results of broader scope stemmed during the various formalizations, this first article stays at a very generic level, with results and registrations about objects already in the Mizar Mathematical Library. \par Without introducing the Language structure yet, three fundamental definitions of wide applicability are also given: the `unambiguous' attribute (see \cite{lothaire2002algebraic}, definition on page 5), the functor `-multiCat', which is the iteration of `\^\,' over a FinSequence of FinSequence, and the functor SubstWith, which realizes the substitution of a single symbol inside a generic FinSequence.

- Marco B. Caminati.
Definition of First Order Language with Arbitrary Alphabet. {S}yntax of Terms, Atomic Formulas and their Subterms,
Formalized Mathematics 19(3),
pages 169-178, 2011. MML Identifier:
FOMODEL1

**Summary**: Second of a series of articles laying down the bases for classical first order model theory. A language is defined basically as a tuple made of an integer-valued function (adicity), a symbol of equality and a symbol for the NOR logical connective. The only requests for this tuple to be a language is that the value of the adicity in = is -2 and that its preimage (i.e. the variables set) in 0 is infinite. Existential quantification will be rendered (see \cite{FOMODEL2.ABS}) by mere prefixing a formula with a letter. Then the hierarchy among symbols according to their adicity is introduced, taking advantage of attributes and clusters.\par The strings of symbols of a language are depth-recursively classified as terms using the standard approach (see for example \cite{pohlers1992introduction}, definition 1.1.2); technically, this is done here by deploying the `-multiCat' functor and the `unambiguous' attribute previously introduced in \cite{FOMODEL0.ABS}, and the set of atomic formulas is introduced. The set of all terms is shown to be unambiguous with respect to concatenation; we say that it is a prefix set. This fact is exploited to uniquely define the subterms both of a term and of an atomic formula without resorting to a parse tree.

- Marco B. Caminati.
First Order Languages: Further Syntax and Semantics,
Formalized Mathematics 19(3),
pages 179-192, 2011. MML Identifier:
FOMODEL2

**Summary**: Third of a series of articles laying down the bases for classical first order model theory. Interpretation of a language in a universe set. Evaluation of a term in a universe. Truth evaluation of an atomic formula. Reassigning the value of a symbol in a given interpretation. Syntax and semantics of a non atomic formula are then defined concurrently (this point is explained in \cite{caminati2010basic}, 4.2.1). As a consequence, the evaluation of any w.f.f. string and the relation of logical implication are introduced. Depth of a formula. Definition of satisfaction and entailment (aka entailment or logical implication) relations, see \cite{ebbinghaus1994mathematical} III.3.2 and III.4.1 respectively.

- Marco B. Caminati.
Free Interpretation, Quotient Interpretation and Substitution of a Letter with a Term for First Order Languages,
Formalized Mathematics 19(3),
pages 193-203, 2011. MML Identifier:
FOMODEL3

**Summary**: Fourth of a series of articles laying down the bases for classical first order model theory. This paper supplies a toolkit of constructions to work with languages and interpretations, and results relating them. The free interpretation of a language, having as a universe the set of terms of the language itself, is defined.\par The quotient of an interpreteation with respect to an equivalence relation is built, and shown to remain an interpretation when the relation respects it. Both the concepts of quotient and of respecting relation are defined in broadest terms, with respect to objects as general as possible.\par Along with the trivial symbol substitution generally defined in \cite{FOMODEL0.ABS}, the more complex substitution of a letter with a term is defined, basing right on the free interpretation just introduced, which is a novel approach, to the author's knowledge. A first important result shown is that the quotient operation commute in some sense with term evaluation and reassignment functors, both introduced in \cite{FOMODEL2.ABS} (theorem 3, theorem 15). A second result proved is substitution lemma (theorem 10, corresponding to III.8.3 of \cite{ebbinghaus1994mathematical}). This will be vital for proving satisfiability theorem and correctness of a certain sequent derivation rule in \cite{FOMODEL4.ABS}. A third result supplied is that if two given languages coincide on the letters of a given FinSequence, their evaluation of it will also coincide. This too will be instrumental in \cite{FOMODEL4.ABS} for proving correctness of another rule. Also, the Depth functor is shown to be invariant with respect to term substitution in a formula.

- Marco B. Caminati.
Sequent Calculus, Derivability, Provability. {G}\"odel's Completeness Theorem,
Formalized Mathematics 19(3),
pages 205-222, 2011. MML Identifier:
FOMODEL4

**Summary**: Fifth of a series of articles laying down the bases for classical first order model theory. This paper presents multiple themes: first it introduces sequents, rules and sets of rules for a first order language L as L-dependent types. Then defines derivability and provability according to a set of rules, and gives several technical lemmas binding all those concepts. Following that, it introduces a fixed set D of derivation rules, and proceeds to convert them to Mizar functorial cluster registrations to give the user a slick interface to apply them.\par The remaining goals summon all the definitions and results introduced in this series of articles. First: D is shown to be correct and having the requisites to deliver a sensible definition of Henkin model (see \cite{ebbinghaus1994mathematical}). Second: as a particular application of all the machinery built thus far, the satisfiability and G\"odel completeness theorems are shown when restricting to countable languages. The techniques used to attain this are inspired from \cite{ebbinghaus1994mathematical}, then heavily modified with the twofold goal of embedding them into the more flexible framework of a variable ruleset here introduced, and of proving completeness of a set of rules more sparing than the one there used; in particular the simpler ruleset allowed to avoid the definition and tractation of free occurence of a literal, a fact which, along with shortening proofs, is remarkable in its own right. A preparatory account of some of the ideas used in the proofs given here can be found in \cite{caminati2009yet}.

- Artur Korni{\l}owicz.
{M}azur-{U}lam Theorem,
Formalized Mathematics 19(3),
pages 127-130, 2011. MML Identifier:
MAZURULM
- Number 4
- Artur Korni{\l}owicz.
Cayley's Theorem,
Formalized Mathematics 19(4),
pages 223-225, 2011. MML Identifier:
CAYLEY

**Summary**: The article formalizes the Cayley's theorem saying that every group $G$ is isomorphic to a subgroup of the symmetric group on $G$.

- Peter Jaeger.
{B}orel-{C}antelli Lemma,
Formalized Mathematics 19(4),
pages 227-232, 2011. MML Identifier:
BOR_CANT

**Summary**: This article is about the Borel-Cantelli Lemma in probability theory. Necessary definitions and theorems are given in \cite{klenke:2006} and \cite{georgii:2004}.

- Keiko Narita, Artur Korni{\l}owicz, Yasunari Shidama.
More on the Continuity of Real Functions,
Formalized Mathematics 19(4),
pages 233-239, 2011. MML Identifier:
NFCONT_4

**Summary**: In this article we demonstrate basic properties of the continuous functions from $\mathbb{R}$ to ${\cal R}^{n}$ which correspond to state space equations in control engineering.

- Grzegorz Bancerek.
Representation Theorem for Stacks,
Formalized Mathematics 19(4),
pages 241-250, 2011. MML Identifier:
STACKS_1

**Summary**: In the paper the concept of stacks is formalized. As the main result the Theorem of Representation for Stacks is given. Formalization is done according to \cite{Salwicki}.

- Artur Korni{\l}owicz.
Cayley's Theorem,
Formalized Mathematics 19(4),
pages 223-225, 2011. MML Identifier:
CAYLEY
**Volume 20, 2012**- Number 1
- Peter Jaeger.
{E}lementary Introduction to Stochastic Finance in Discrete Time,
Formalized Mathematics 20(1),
pages 1-5, 2012. MML Identifier:
FINANCE1

**Summary**: This article gives an elementary introduction to stochastic finance (in discrete time). A formalization of random variables is given and some elements of Borel sets are considered. Furthermore, special functions (for buying a present portfolio and the value of a portfolio in the future) and some statements about the relation between these functions are introduced. For details see: \cite{klenke:2006} (p. 185), \cite{georgii:2004} (pp. 12, 20), \cite{follmerschied:2004} (pp. 3--6).

- Grzegorz Bancerek, Hidetsune Kobayashi, Artur Korni{\l}owicz.
{V}aluation Theory. {P}art {I},
Formalized Mathematics 20(1),
pages 7-14, 2012. MML Identifier:
FVALUAT1

**Summary**: In the article we introduce a valuation function over a field \cite{EmilArtin}. Ring of non negative elements and its ideal of positive elements have been also defined.

- Katuhiko Kanazashi, Hiroyuki Okazaki, Yasunari Shidama.
{F}unctional Space ${\bmC}(\Omega)$, ${\bmCz(\Omega)}$,
Formalized Mathematics 20(1),
pages 15-22, 2012. MML Identifier:
CC0SP2

**Summary**: In this article, first we give a definition of a functional space which is constructed from all complex-valued continuous functions defined on a compact topological space. We prove that this functional space is a Banach algebra. Next, we give a definition of a function space which is constructed from all complex-valued continuous functions with bounded support. We also prove that this function space is a complex normed space.

- Karol P\k{a}k.
{T}he Rotation Group,
Formalized Mathematics 20(1),
pages 23-29, 2012. MML Identifier:
MATRTOP3

**Summary**: We introduce length-preserving linear transformations of Euclidean topological spaces. We also introduce rotation which preserves orientation (proper rotation) and reverses orientation (improper rotation). We show that every rotation that preserves orientation can be represented as a composition of base proper rotations. And finally, we show that every rotation that reverses orientation can be represented as a composition of proper rotations and one improper rotation.

- Yasunari Shidama.
{D}ifferentiable Functions on Normed Linear Spaces,
Formalized Mathematics 20(1),
pages 31-40, 2012. MML Identifier:
NDIFF_5

**Summary**: In this article, we formalize differentiability of functions on normed linear spaces. Partial derivative, mean value theorem for vector-valued functions, continuous differentiability, etc. are formalized. As it is well known, there is no exact analog of the mean value theorem for vector-valued functions. However a certain type of generalization of the mean value theorem for vector-valued functions is obtained as follows: If $||f'(x + t\cdot h)||$ is bounded for $t$ between 0 and 1 by some constant $M,$ then $|| f(x + t\cdot h) - f(x)|| \leq M \cdot ||h||.$ This theorem is called the mean value theorem for vector-valued functions. By this theorem, the relation between the (total) derivative and the partial derivatives of a function is derived \cite{Schwartz:1967}.

- Marco Riccardi.
{P}lanes and Spheres as Topological
Manifolds. {S}tereographic Projection,
Formalized Mathematics 20(1),
pages 41-45, 2012. MML Identifier:
MFOLD_2

**Summary**: The goal of this article is to show some examples of topological manifolds: planes and spheres in Euclidean space. In doing it, the article introduces the stereographic projection \cite{JLee2000}.

- Yuichi Futa, Hiroyuki Okazaki, Yasunari Shidama.
{$\mathbb Z$}-modules,
Formalized Mathematics 20(1),
pages 47-59, 2012. MML Identifier:
ZMODUL01

**Summary**: In this article, we formalize $\mathbb Z$-module, that is a module over integer ring. $\mathbb Z$-module is necessary for lattice problems, LLL (Lenstra-Lenstra-Lov\'{a}sz) base reduction algorithm and cryptographic systems with lattices {\cite{Micci:2002}}.

- Hiroshi Yamazaki, Czes{\l}aw Byli{\'n}ski, Katsumi Wasaki.
{M}orphology for Image Processing. {P}art {I},
Formalized Mathematics 20(1),
pages 61-63, 2012. MML Identifier:
MORPH_01

**Summary**: In this article we defined mathematical morphology image processing with set operations. First, we defined Minkowski set operations and proved their properties. Next, we defined basic image processing, dilation and erosion proving basic fact about them \cite{Heijmans:1994}, \cite{Soille:2003}.

- Keiko Narita, Artur Korni{\l}owicz, Yasunari Shidama.
{T}he Differentiable Functions from $\mathbb{R}$ into ${\cal{R}}^n$,
Formalized Mathematics 20(1),
pages 65-71, 2012. MML Identifier:
NDIFF_4

**Summary**: In control engineering, differentiable partial functions from $\mathbb{R}$ into ${\cal{R}}^n$ play a very important role. In this article, we formalized basic properties of such functions.

- Xiquan Liang, Tao Wang.
{S}ome Basic Properties of Some Special Matrices. {P}art {III},
Formalized Mathematics 20(1),
pages 73-77, 2012. MML Identifier:
MATRIX17

**Summary**: This article describes definitions of subsymmetric matrix, anti-subsymmetric matrix, central symmetric matrix, symmetry circulant matrix and their basic properties.

- Keiichi Miyajima, Artur Korni{\l}owicz, Yasunari Shidama.
{R}iemann Integral of Functions
from $\mathbb{R}$ into $n$-dimensional Real Normed Space,
Formalized Mathematics 20(1),
pages 79-86, 2012. MML Identifier:
INTEGR19

**Summary**: In this article, we define the Riemann integral on functions $\mathbb R$ into $n$-dimensional real normed space and prove the linearity of this operator. As a result, the Riemann integration can be applied to the wider range. Our method refers to the \cite{Murray:1974}.

- Yuichi Futa, Hiroyuki Okazaki, Daichi Mizushima, Yasunari Shidama.
{O}perations of Points on Elliptic Curve in Projective Coordinate,
Formalized Mathematics 20(1),
pages 87-95, 2012. MML Identifier:
EC_PF_2

**Summary**: In this article, we formalize operations of points on an elliptic curve over $\bf{GF}(p)$. Elliptic curve cryptography \cite{BSS99}, whose security is based on a difficulty of discrete logarithm problem of elliptic curves, is important for information security. We prove that the two operations of points: $\mbox{\tt compell}_\mbox{\tt ProjCo}$ and $\mbox{\tt addell}_\mbox{\tt ProjCo}$ are unary and binary operations of a point over the elliptic curve.

- Peter Jaeger.
{E}lementary Introduction to Stochastic Finance in Discrete Time,
Formalized Mathematics 20(1),
pages 1-5, 2012. MML Identifier:
FINANCE1
- Number 2
- Marco Riccardi, Artur Korni{\l}owicz.
{F}undamental Group of $n$-sphere for $n \geq 2$,
Formalized Mathematics 20(2),
pages 97-104, 2012. MML Identifier:
TOPALG_6

**Summary**: Triviality of fundamental groups of spheres of dimension greater than 1 is proven, \cite{JLee2000}.

- Artur Korni{\l}owicz, Marco Riccardi.
{T}he {B}orsuk-{U}lam Theorem,
Formalized Mathematics 20(2),
pages 105-112, 2012. MML Identifier:
BORSUK_7

**Summary**: The Borsuk-Ulam theorem about antipodals is proven, \cite[pp.~32--33]{Hatcher}.

- Noboru Endou, Hiroyuki Okazaki, Yasunari Shidama.
{H}igher-Order Partial Differentiation,
Formalized Mathematics 20(2),
pages 113-124, 2012. MML Identifier:
PDIFF_9

**Summary**: In this article, we shall extend the formalization of \cite{PDIFF_1.ABS} to discuss higher-order partial differentiation of real valued functions. The linearity of this operator is also proved (refer to \cite{PDIFF_1.ABS}, \cite{PDIFF_6.ABS} and \cite{PDIFF_7.ABS} for partial differentiation).

- Hiroyuki Okazaki, Yasunari Shidama.
{F}ormalization of the Data Encryption Standard,
Formalized Mathematics 20(2),
pages 125-146, 2012. MML Identifier:
DESCIP_1

**Summary**: In this article we formalize DES (the Data Encryption Standard), that was the most widely used symmetric cryptosystem in the world. DES is a block cipher which was selected by the National Bureau of Standards as an official Federal Information Processing Standard for the United States in 1976~\cite{FIPS}.

- Grzegorz Bancerek.
{S}emantics of {MML} Query,
Formalized Mathematics 20(2),
pages 147-155, 2012. MML Identifier:
MMLQUERY

**Summary**: In the paper the semantics of MML Query queries is given. The formalization is done according to \cite{Bancerek2006}.

- Boris A. Shminke.
{R}outh's, {M}enelaus' and Generalized {C}eva's Theorems,
Formalized Mathematics 20(2),
pages 157-159, 2012. MML Identifier:
MENELAUS

**Summary**: The goal of this article is to formalize Ceva's theorem that is in the \cite{Freek-100-theorems} on the web. Alongside with it formalizations of Routh's, Menelaus' and generalized form of Ceva's theorem itself are provided.

- Piotr Rudnicki, Lorna Stewart.
{S}imple Graphs as Simplicial Complexes: the {M}ycielskian of a Graph,
Formalized Mathematics 20(2),
pages 161-174, 2012. MML Identifier:
SCMYCIEL

**Summary**: Harary~\cite[p.~7]{Harary69} claims that Veblen~\cite[p.~2]{Veblen31} first suggested to formalize simple graphs using simplicial complexes. We have developed basic terminology for simple graphs as at most 1-dimensional complexes.\par We formalize this new setting and then reprove Mycielski's~\cite{Mycielski55} construction resulting in a triangle-free graph with arbitrarily large chromatic number. A different formalization of similar material is in~\cite{MYCIELSK.ABS}.

- Hiroyuki Okazaki, Yosiki Aoki, Yasunari Shidama.
{E}xtended {E}uclidean Algorithm and {CRT} Algorithm,
Formalized Mathematics 20(2),
pages 175-179, 2012. MML Identifier:
NTALGO_1

**Summary**: In this article we formalize some number theoretical algorithms, Euclidean Algorithm and Extended Euclidean Algorithm \cite{Knuth2}. Besides the $a$ gcd $b$, Extended Euclidean Algorithm can calculate a pair of two integers $(x,y)$ that holds $ax+by = a$ gcd $b$. In addition, we formalize an algorithm that can compute a solution of the Chinese remainder theorem by using Extended Euclidean Algorithm. Our aim is to support the implementation of number theoretic tools. Our formalization of those algorithms is based on the source code of the NZMATH, a number theory oriented calculation system developed by Tokyo Metropolitan University \cite{NZMATH}.

- Christoph Schwarzweller.
{I}ntroduction to Rational Functions,
Formalized Mathematics 20(2),
pages 181-191, 2012. MML Identifier:
RATFUNC1

**Summary**: In this article we formalize rational functions as pairs of polynomials and define some basic notions including the degree and evaluation of rational functions \cite{HH90}. The main goal of the article is to provide properties of rational functions necessary to prove a theorem on the stability of networks.

- Marco Riccardi, Artur Korni{\l}owicz.
{F}undamental Group of $n$-sphere for $n \geq 2$,
Formalized Mathematics 20(2),
pages 97-104, 2012. MML Identifier:
TOPALG_6
- Number 3
- Julian J. Schl\"oder, Peter Koepke.
{T}ransition of Consistency and Satisfiability
under Language Extensions,
Formalized Mathematics 20(3),
pages 193-197, 2012. MML Identifier:
QC_TRANS

**Summary**: This article is the first in a series of two Mizar articles constituting a formal proof of the G\"odel Completeness theorem \cite{Goedel:1930} for uncountably large languages. We follow the proof given in \cite{Ebbinghaus2007}. The present article contains the techniques required to expand formal languages. We prove that consistent or satisfiable theories retain these properties under changes to the language they are formulated in.

- Julian J. Schl\"oder, Peter Koepke.
{T}he {G}\"odel Completeness Theorem for Uncountable Languages,
Formalized Mathematics 20(3),
pages 199-203, 2012. MML Identifier:
GOEDCPUC

**Summary**: This article is the second in a series of two Mizar articles constituting a formal proof of the G\"odel Completeness theorem \cite{Goedel:1930} for uncountably large languages. We follow the proof given in \cite{Ebbinghaus2007}. The present article contains the techniques required to expand a theory such that the expanded theory contains witnesses and is negation faithful. Then the completeness theorem follows immediately.

- Yuichi Futa, Hiroyuki Okazaki, Yasunari Shidama.
{Q}uotient Module of {$\mathbb Z$}-module,
Formalized Mathematics 20(3),
pages 205-214, 2012. MML Identifier:
ZMODUL02

**Summary**: In this article we formalize a quotient module of $\mathbb Z$-module and a vector space constructed by the quotient module. We formally prove that for a $\mathbb Z$-module $V$ and a prime number $p$, a quotient module $V / p V$ has the structure of a vector space over $\mathbb{F}_p$. $\mathbb Z$-module is necessary for lattice problems, LLL (Lenstra, Lenstra and Lov\'asz) base reduction algorithm and cryptographic systems with lattices \cite{Micci:2002}. Some theorems in this article are described by translating theorems in \cite{RLVECT_2.ABS} and \cite{RLVECT_3.ABS} into theorems of $\mathbb Z$-module.

- Mariusz Giero.
{T}he Derivations of Temporal Logic Formulas,
Formalized Mathematics 20(3),
pages 215-219, 2012. MML Identifier:
LTLAXIO2

**Summary**: This is a preliminary article to prove the completeness theorem of an extension of basic propositional temporal logic. We base it on the proof of completeness for basic propositional temporal logic given in \cite{KroMer}. We introduce $n$-ary connectives and prove their properties. We derive temporal logic formulas.

- Mariusz Giero.
{T}he Properties of Sets of Temporal Logic Subformulas,
Formalized Mathematics 20(3),
pages 221-226, 2012. MML Identifier:
LTLAXIO3

**Summary**: This is a second preliminary article to prove the completeness theorem of an extension of basic propositional temporal logic. We base it on the proof of completeness for basic propositional temporal logic given in \cite{KroMer}. We introduce two modified definitions of a subformula. In the former one we treat until-formula as indivisible. In the latter one, we extend the set of subformulas of until-formulas by a special disjunctive formula. This is needed to construct a temporal model. We also define an ordered positive-negative pair of finite sequences of formulas (PNP). PNPs represent states of a temporal model.

- Mariusz Giero.
{W}eak Completeness Theorem for Propositional Linear Time Temporal Logic,
Formalized Mathematics 20(3),
pages 227-234, 2012. MML Identifier:
LTLAXIO4

**Summary**: We prove weak (finite set of premises) completeness theorem for extended propositional linear time temporal logic with irreflexive version of until-operator. We base it on the proof of completeness for basic propositional linear time temporal logic given in \cite{KroMer} which roughly follows the idea of the Henkin-Hasenjaeger method for classical logic. We show that a temporal model exists for every formula which negation is not derivable (Satisfiability Theorem). The contrapositive of that theorem leads to derivability of every valid formula. We build a tree of consistent and complete~PNPs which is used to construct the model.

- Karol P\k{a}k.
{T}he Friendship Theorem,
Formalized Mathematics 20(3),
pages 235-237, 2012. MML Identifier:
FRIENDS1

**Summary**: In this article we prove the friendship theorem according to the article \cite{MAlbert}, which states that if a group of people has the property that any pair of persons have exactly one common friend, then there is a universal friend, i.e. a person who is a friend of every other person in the group.

- Grzegorz Bancerek.
{F}ree Term Algebras,
Formalized Mathematics 20(3),
pages 239-256, 2012. MML Identifier:
MSAFREE4

**Summary**: We interoduce a new characterization of algebras of normal forms of term rewriting systems \cite{KlopTRS} as algerbras of term free in itself (any function from free generators into the algebra generates endomorphism of the algebra). Introduced algebras are free in classes of algebras satisfying some sets of equalities. Their universes are subsets of all terms and the denotations of operation symbols are partially identical with the operations of construction of terms. These algebras are compiler algebras requiring some equalities of terms, e.g., associativity of addition.

- Julian J. Schl\"oder, Peter Koepke.
{T}ransition of Consistency and Satisfiability
under Language Extensions,
Formalized Mathematics 20(3),
pages 193-197, 2012. MML Identifier:
QC_TRANS
- Number 4
- Hiroyuki Okazaki.
{P}osterior Probability on Finite Set,
Formalized Mathematics 20(4),
pages 257-263, 2012. MML Identifier:
DIST_2

**Summary**: In \cite{DIST_1.ABS} we formalized probability and probability distribution on a finite sample space. In this article first we propose a formalization of the class of finite sample spaces whose element's probability distributions are equivalent with each other. Next, we formalize the probability measure of the class of sample spaces we have formalized above. Finally, we formalize the sampling and posterior probability.

- Na Ma, Xiquan Liang.
{B}asic Properties of Primitive Root and Order Function,
Formalized Mathematics 20(4),
pages 265-269, 2012. MML Identifier:
INT_8

**Summary**: In this paper we defined the reduced residue system and proved its fundamental properties. Then we proved the basic properties of the order function. Finally, we defined the primitive root and proved its fundamental properties. Our work is based on \cite{HuaLooKeng:1957}, \cite{Dexin:1965}, and \cite{HardyWright2007}.

- Hideki Sakurai, Hiroyuki Okazaki, Yasunari Shidama.
{B}anach's Continuous Inverse Theorem and Closed Graph Theorem,
Formalized Mathematics 20(4),
pages 271-274, 2012. MML Identifier:
LOPBAN_7

**Summary**: In this article we formalize one of the most important theorems of linear operator theory -- the Closed Graph Theorem commonly used in a standard text book such as \cite{miyadera:1972} in Chapter 24.3. It states that a surjective closed linear operator between Banach spaces is bounded.

- Yuichi Futa, Hiroyuki Okazaki, Yasunari Shidama.
{F}ree {$\mathbb Z$}-module,
Formalized Mathematics 20(4),
pages 275-280, 2012. MML Identifier:
ZMODUL03

**Summary**: In this article we formalize a free $\mathbb Z$-module and its rank. We formally prove that for a free finite rank $\mathbb Z$-module $V$, the number of elements in its basis, that is a rank of the $\mathbb Z$-module, is constant regardless of the selection of its basis. $\mathbb Z$-module is necessary for lattice problems, LLL(Lenstra, Lenstra and Lov\'asz) base reduction algorithm and cryptographic systems with lattice \cite{Micci:2002}. Some theorems in this article are described by translating theorems in \cite{RLVECT_3.ABS} and \cite{RLVECT_5.ABS} into theorems of $\mathbb Z$-module.

- Artur Korni{\l}owicz.
{C}ayley-{D}ickson Construction,
Formalized Mathematics 20(4),
pages 281-290, 2012. MML Identifier:
CAYLDICK

**Summary**: Cayley-Dickson construction produces a~sequence of normed algebras over real numbers. Its consequent applications result in complex numbers, quaternions, octonions, etc. In this paper we formalize the construction and prove its basic properties.

- Keiichi Miyajima, Artur Korni{\l}owicz, Yasunari Shidama.
{C}ontracting Mapping on Normed Linear Space,
Formalized Mathematics 20(4),
pages 291-301, 2012. MML Identifier:
ORDEQ_01

**Summary**: In this article, we described the contracting mapping on normed linear space. Furthermore, we applied that mapping to ordinary differential equations on real normed space. Our method is based on the one presented by Schwartz \cite{Schwartz:1967:2:5}.

- Artur Korni{\l}owicz.
{P}roducts in Categories without Uniqueness of {\bf cod} and {\bf dom},
Formalized Mathematics 20(4),
pages 303-307, 2012. MML Identifier:
ALTCAT_5

**Summary**: The paper introduces Cartesian products in categories without uniqueness of {\bf cod} and {\bf dom}. It is proven that set-theoretical product is the product in the category Ens \cite{SEMAD}.

- Grzegorz Bancerek.
{P}rogram Algebra over an Algebra,
Formalized Mathematics 20(4),
pages 309-341, 2012. MML Identifier:
AOFA_A00

**Summary**: We introduce an algebra with free variables, an algebra with undefined values, a program algebra over a term algebra, an algebra with integers, and an algebra with arrays. Program algebra is defined as universal algebra with assignments. Programs depend on the set of generators with supporting variables and supporting terms which determine the value of free variables in the next state. The execution of a program is changing state according to successor function using supporting terms.

- Kenichi Arai, Hiroyuki Okazaki, Yasunari Shidama.
{I}somorphisms of Direct Products of Finite Cyclic Groups,
Formalized Mathematics 20(4),
pages 343-347, 2012. MML Identifier:
GROUP_14

**Summary**: In this article, we formalize that every finite cyclic group is isomorphic to a direct product of finite cyclic groups which orders are relatively prime. This theorem is closely related to the Chinese Remainder theorem (\cite{INT_6.ABS}) and is a useful lemma to prove the basis theorem for finite abelian groups and the fundamental theorem of finite abelian groups. Moreover, we formalize some facts about the product of a finite sequence of abelian groups.

- Yasushige Watase, Noboru Endou, Yasunari Shidama.
{O}n $L^1$ Space Formed by Complex-Valued Partial Functions,
Formalized Mathematics 20(4),
pages 349-357, 2012. MML Identifier:
LPSPACC1

**Summary**: In this article, we formalized $L^1$ space formed by complex-valued partial functions \cite{Halmos74}, \cite{RUDIN:2}. The real-valued case was formalized in \cite{LPSPACE1.ABS} and this article is its generalization.

- Hiroyuki Okazaki.
{P}osterior Probability on Finite Set,
Formalized Mathematics 20(4),
pages 257-263, 2012. MML Identifier:
DIST_2
**Volume 21, 2013**- Number 1
- Grzegorz Bancerek.
{A}nalysis of Algorithms: An Example of a Sort Algorithm,
Formalized Mathematics 21(1),
pages 1-23, 2013. MML Identifier:
AOFA_A01

**Summary**: We analyse three algorithms: exponentiation by squaring, calculation of maximum, and sorting by exchanging in terms of program algebra over an algebra.

- Katuhiko Kanazashi, Hiroyuki Okazaki, Yasunari Shidama.
{T}he {$C^k$} Space,
Formalized Mathematics 21(1),
pages 25-31, 2013. MML Identifier:
CKSPACE1

**Summary**: In this article, we formalize continuous differentiability of real-valued functions on $n$-dimensional real normed linear spaces. Next, we give a definition of the $C^k$ space according to \cite{Kosaku:1996}.

- Hiroyuki Okazaki, Yasunari Shidama.
{R}andom Variables and Product of Probability Spaces,
Formalized Mathematics 21(1),
pages 33-39, 2013. MML Identifier:
RANDOM_3

**Summary**: We have been working on the formalization of the probability and the randomness. In \cite{RANDOM_1.ABS} and \cite{RANDOM_2.ABS}, we formalized some theorems concerning the real-valued random variables and the product of two probability spaces. In this article, we present the generalized formalization of \cite{RANDOM_1.ABS} and \cite{RANDOM_2.ABS}. First, we formalize the random variables of arbitrary set and prove the equivalence between random variable on $\Sigma$, Borel sets and a real-valued random variable on $\Sigma$. Next, we formalize the product of countably infinite probability spaces.

- Grzegorz Bancerek.
{S}emantics of {MML} {Q}uery - Ordering,
Formalized Mathematics 21(1),
pages 41-46, 2013. MML Identifier:
MMLQUER2

**Summary**: Semantics of order directives of MML Query is presented. The formalization is done according to \cite{Bancerek2006}.

- Agnieszka Rowi\'nska-Schwarzweller, Christoph Schwarzweller.
{A} Test for the Stability of Networks,
Formalized Mathematics 21(1),
pages 47-54, 2013. MML Identifier:
HURWITZ2

**Summary**: A complex polynomial is called a Hurwitz polynomial, if all its roots have a real part smaller than zero. This kind of polynomial plays an all-dominant role in stability checks of electrical (analog or digital) networks.\\ In this article we prove that a polynomial $p$ can be shown to be Hurwitz by checking whether the rational function $e(p)/o(p)$ can be realized as a reactance of one port, that is as an electrical impedance or admittance consisting of inductors and capacitors. Here $e(p)$ and $o(p)$ denote the even and the odd part of $p$ \cite{Unb93}.

- Adam Grabowski.
{R}elational Formal Characterization of Rough Sets,
Formalized Mathematics 21(1),
pages 55-64, 2013. MML Identifier:
ROUGHS_2

**Summary**: The notion of a rough set, developed by Pawlak \cite{Pawlak1982}, is an important tool to describe situation of incomplete or partially unknown information. In this article, which is essentially the continuation of \cite{ROUGHS_1.ABS}, we try to give the characterization of approximation operators in terms of ordinary properties of underlying relations (some of them, as serial and mediate relations, were not available in the Mizar Mathematical Library). Here we drop the classical equivalence- and tolerance-based models of rough sets \cite{SkowronS96} trying to formalize some parts of \cite{Zhu:2007} following also \cite{Yao96} in some sense (Propositions 1--8, Corr. 1 and 2; the complete description is available in the Mizar script). Our main problem was that informally, there is a direct correspondence between relations and underlying properties, in our approach however \cite{GrabowskiJ10}, which uses relational structures rather than relations, we had to switch between classical (based on pure set theory) and abstract (using the notion of a structure) parts of the Mizar Mathematical Library. Our next step will be translation of these properties into the pure language of Mizar attributes.

- Hiroyuki Okazaki, Hiroshi Yamazaki, Yasunari Shidama.
{I}somorphisms of Direct Products of Finite Commutative Groups,
Formalized Mathematics 21(1),
pages 65-74, 2013. MML Identifier:
GROUP_17

**Summary**: We have been working on the formalization of groups. In \cite{GROUP_14.ABS}, we encoded some theorems concerning the product of cyclic groups. In this article, we present the generalized formalization of \cite{GROUP_14.ABS}. First, we show that every finite commutative group which order is composite number is isomorphic to a direct product of finite commutative groups which orders are relatively prime. Next, we describe finite direct products of finite commutative groups.

- Grzegorz Bancerek.
{A}nalysis of Algorithms: An Example of a Sort Algorithm,
Formalized Mathematics 21(1),
pages 1-23, 2013. MML Identifier:
AOFA_A01
- Number 2
- Kenichi Arai, Hiroyuki Okazaki.
{$N$}-Dimensional Binary Vector Spaces,
Formalized Mathematics 21(2),
pages 75-81, 2013. MML Identifier:
NBVECTSP

**Summary**: The binary set $\{0,1\}$ together with modulo-2 addition and multiplication is called a binary field, which is denoted by $\mathbb{F}_2$. The binary field $\mathbb{F}_2$ is defined in \cite{BSPACE.ABS}. A vector space over $\mathbb{F}_2$ is called a binary vector space. The set of all binary vectors of length $n$ forms an $n$-dimensional vector space $V_n$ over $\mathbb{F}_2$. Binary fields and $n$-dimensional binary vector spaces play an important role in practical computer science, for example, coding theory \cite{ECC:2006} and cryptology. In cryptology, binary fields and $n$-dimensional binary vector spaces are very important in proving the security of cryptographic systems \cite{Lai1994}. In this article we define the $n$-dimensional binary vector space $V_n$. Moreover, we formalize some facts about the $n$-dimensional binary vector space $V_n$.

- Adam St. Arnaud, Piotr Rudnicki.
{S}ome Properties of the {S}orgenfrey Line and the {S}orgenfrey Plane,
Formalized Mathematics 21(2),
pages 83-85, 2013. MML Identifier:
TOPGEN_6

**Summary**: We first provide a modified version of the proof in \cite{TOPGEN_5.ABS} that the Sorgenfrey line is $T_1$. Here, we prove that it is in fact $T_2$, a stronger result. Next, we prove that all subspaces of ${\mathbb R}^{\bf 1}$ (that is the real line with the usual topology) are Lindel\"of. We utilize this result in the proof that the Sorgenfrey line is Lindel\"of, which is based on the proof found in \cite{Engelking:1968}. Next, we construct the Sorgenfrey plane, as the product topology of the Sorgenfrey line and itself. We prove that the Sorgenfrey plane is not Lindel\"of, and therefore the product space of two Lindel\"of spaces need not be Lindel\"of. Further, we note that the Sorgenfrey line is regular, following from \cite{TOPGEN_5.ABS}:59. Next, we observe that the Sorgenfrey line is normal since it is both regular and Lindel\"of. Finally, we prove that the Sorgenfrey plane is not normal, and hence the product of two normal spaces need not be normal. The proof that the Sorgenfrey plane is not normal and many of the lemmas leading up to this result are modelled after the proof in \cite{TOPGEN_5.ABS}, that the Niemytzki plane is not normal. Information was also gathered from \cite{SteenSeebach:1978}.

- Adam Naumowicz, Rados{\l}aw Piliszek.
{M}ore on Divisibility Criteria for Selected Primes,
Formalized Mathematics 21(2),
pages 87-94, 2013. MML Identifier:
NUMERAL2

**Summary**: This paper is a continuation of \cite{NUMERAL1.ABS}, where the divisibility criteria for initial prime numbers based on their representation in the decimal system were formalized. In the current paper we consider all primes up to 101 to demonstrate the method presented in \cite{Briggs2000}.

- Noboru Endou, Yasunari Shidama.
{D}ifferentiation in Normed Spaces,
Formalized Mathematics 21(2),
pages 95-102, 2013. MML Identifier:
NDIFF_6

**Summary**: In this article we formalized the Fr\'echet differentiation. It is defined as a generalization of the differentiation of a real-valued function of a single real variable to more general functions whose domain and range are subsets of normed spaces \cite{Schwartz:1981}.

- Adam Grabowski.
{P}olygonal Numbers,
Formalized Mathematics 21(2),
pages 103-113, 2013. MML Identifier:
NUMPOLY1

**Summary**: In the article the formal characterization of triangular numbers (famous from \cite{Gauss:1986} and words ``EYPHKA! num = $\Delta + \Delta + \Delta$") \cite{Heath:1921} is given. Our primary aim was to formalize one of the items (\#42) from Wiedijk's Top 100 Mathematical Theorems list \cite{Freek-100-theorems}, namely that the sequence of sums of reciprocals of triangular numbers converges to 2. This Mizar representation was written in 2007. As the Mizar language evolved and attributes with arguments were implemented, we decided to extend these lines and we characterized polygonal numbers.\par We formalized centered polygonal numbers, the connection between triangular and square numbers, and also some equalities involving Mersenne primes and perfect numbers. We gave also explicit formula to obtain from the polygonal number its ordinal index. Also selected congruences modulo 10 were enumerated. Our work basically covers the Wikipedia item for triangular numbers and the Online Encyclopedia of Integer Sequences (\url{http://oeis.org/A000217}). \par An interesting related result \cite{Guy:1994} could be the proof of Lagrange's four-square theorem or Fermat's polygonal number theorem \cite{Weil:1983}.

- Yuichi Futa, Hiroyuki Okazaki, Daichi Mizushima, Yasunari Shidama.
{G}aussian Integers,
Formalized Mathematics 21(2),
pages 115-125, 2013. MML Identifier:
GAUSSINT

**Summary**: Gaussian integer is one of basic algebraic integers. In this article we formalize some definitions about Gaussian integers \cite{Weil:1979}. We also formalize ring (called Gaussian integer ring), $\mathbb Z$-module and $\mathbb Z$-algebra generated by Gaussian integer mentioned above. Moreover, we formalize some definitions about Gaussian rational numbers and Gaussian rational number field. Then we prove that the Gaussian rational number field and a quotient field of the Gaussian integer ring are isomorphic.

- Artur Korni{\l}owicz.
{C}ommutativeness of Fundamental Groups of Topological Groups,
Formalized Mathematics 21(2),
pages 127-131, 2013. MML Identifier:
TOPALG_7

**Summary**: In this article we prove that fundamental groups based at the unit point of topological groups are commutative \cite{Hatcher}.

- Hiroyuki Okazaki, Yuichi Futa, Yasunari Shidama.
{C}onstructing Binary {H}uffman Tree,
Formalized Mathematics 21(2),
pages 133-143, 2013. MML Identifier:
HUFFMAN1

**Summary**: Huffman coding is one of a most famous entropy encoding methods for lossless data compression \cite{HUFFMAN:1952}. JPEG and ZIP formats employ variants of Huffman encoding as lossless compression algorithms. Huffman coding is a bijective map from source letters into leaves of the Huffman tree constructed by the algorithm. In this article we formalize an algorithm constructing a binary code tree, Huffman tree.

- Keiko Narita, Noboru Endou, Yasunari Shidama.
{R}iemann Integral
of Functions from {$\mathbb R$} into Real {B}anach Space,
Formalized Mathematics 21(2),
pages 145-152, 2013. MML Identifier:
INTEGR20

**Summary**: In this article we deal with the Riemann integral of functions from $\mathbb{R}$ into a real Banach space. The last theorem establishes the integrability of continuous functions on the closed interval of reals. To prove the integrability we defined uniform continuity for functions from $\mathbb{R}$ into a real normed space, and proved related theorems. We also stated some properties of finite sequences of elements of a real normed space and finite sequences of real numbers.\par In addition we proved some theorems about the convergence of sequences. We applied definitions introduced in the previous article \cite{INTEGR18.ABS} to the proof of integrability.

- Adam Grabowski.
{O}n Square-Free Numbers,
Formalized Mathematics 21(2),
pages 153-162, 2013. MML Identifier:
MOEBIUS2

**Summary**: In the article the formal characterization of square-free numbers is shown; in this manner the paper is the continuation of \cite{MOEBIUS1.ABS}. Essentially, we prepared some lemmas for convenient work with numbers (including the proof that the sequence of prime reciprocals diverges \cite{PFTB}) according to \cite{HardyWright} which were absent in the Mizar Mathematical Library. Some of them were expressed in terms of clusters' registrations, enabling automatization machinery available in the Mizar system. Our main result of the article is in the final section; we proved that the lattice of positive divisors of a positive integer $n$ is Boolean if and only if $n$ is square-free.

- Kenichi Arai, Hiroyuki Okazaki.
{$N$}-Dimensional Binary Vector Spaces,
Formalized Mathematics 21(2),
pages 75-81, 2013. MML Identifier:
NBVECTSP
- Number 3
- Noboru Endou, Hiroyuki Okazaki, Yasunari Shidama.
{D}ouble Sequences and Limits,
Formalized Mathematics 21(3),
pages 163-170, 2013. MML Identifier:
DBLSEQ_1

**Summary**: Double sequences are important extension of the ordinary notion of a sequence. In this article we formalized three types of limits of double sequences and the theory of these limits.

- Kenichi Arai, Hiroyuki Okazaki.
{F}ormalization of the Advanced Encryption Standard. {P}art {I},
Formalized Mathematics 21(3),
pages 171-184, 2013. MML Identifier:
AESCIP_1

**Summary**: In this article, we formalize the Advanced Encryption Standard (AES). AES, which is the most widely used symmetric cryptosystem in the world, is a block cipher that was selected by the National Institute of Standards and Technology (NIST) as an official Federal Information Processing Standard for the United States in 2001 \cite{FIPS:197}. AES is the successor to DES \cite{DESCIP_1.ABS}, which was formerly the most widely used symmetric cryptosystem in the world. We formalize the AES algorithm according to \cite{FIPS:197}. We then verify the correctness of the formalized algorithm that the ciphertext encoded by the AES algorithm can be decoded uniquely by the same key. Please note the following points about this formalization: the AES round process is composed of the {\tt SubBytes}, {\tt ShiftRows}, {\tt MixColumns}, and {\tt AddRoundKey} transformations (see \cite{FIPS:197}). In this formalization, the {\tt SubBytes} and {\tt MixColumns} transformations are given as permutations, because it is necessary to treat the finite field GF($2^8$) for those transformations. The formalization of AES that considers the finite field GF($2^8$) is formalized by the future article.

- Keiko Narita, Noboru Endou, Yasunari Shidama.
{T}he Linearity of {R}iemann Integral on Functions
from {$\mathbb R$} into Real {B}anach Space,
Formalized Mathematics 21(3),
pages 185-191, 2013. MML Identifier:
INTEGR21

**Summary**: In this article, we described basic properties of Riemann integral on functions from {$\mathbb R$} into Real Banach Space. We proved mainly the linearity of integral operator about the integral of continuous functions on closed interval of the set of real numbers. These theorems were based on the article \cite{INTEGR19.ABS} and we referred to the former articles about Riemann integral. We applied definitions and theorems introduced in the article \cite{INTEGR18.ABS} and the article \cite{INTEGR20.ABS} to the proof. Using the definition of the article \cite{INTEGR19.ABS}, we also proved some theorems on bounded functions.

- Marco Riccardi.
{O}bject-Free Definition of Categories,
Formalized Mathematics 21(3),
pages 193-205, 2013. MML Identifier:
CAT_6

**Summary**: Category theory was formalized in Mizar with two different approaches \cite{CAT_1.ABS}, \cite{ALTCAT_1.ABS} that correspond to those most commonly used \cite{MacLane:1}, \cite{Borceaux}. Since there is a one-to-one correspondence between objects and identity morphisms, some authors have used an approach that does not refer to objects as elements of the theory, and are usually indicated as object-free category \cite{Adamek:2009} or as arrows-only category \cite{MacLane:1}. In this article is proposed a new definition of an object-free category, introducing the two properties: left composable and right composable, and a simplification of the notation through a symbol, a binary relation between morphisms, that indicates whether the composition is defined. In the final part we define two functions that allow to switch from the two definitions, with and without objects, and it is shown that their composition produces isomorphic categories.

- Hiroshi Yamazaki, Hiroyuki Okazaki, Kazuhisa Nakasho, Yasunari Shidama.
{I}somorphisms of Direct Products of Cyclic Groups
of Prime Power Order,
Formalized Mathematics 21(3),
pages 207-211, 2013. MML Identifier:
GROUP_18

**Summary**: In this paper we formalized some theorems concerning the cyclic groups of prime power order. We formalize that every commutative cyclic group of prime power order is isomorphic to a direct product of family of cyclic groups \cite{GROUP_14.ABS}, \cite{GROUP_17.ABS}.

- Adam Grabowski.
{P}rime Filters and Ideals in Distributive Lattices,
Formalized Mathematics 21(3),
pages 213-221, 2013. MML Identifier:
LATTICEA

**Summary**: The article continues the formalization of the lattice theory (as structures with two binary operations, not in terms of ordering relations). In the Mizar Mathematical Library, there are some attempts to formalize prime ideals and filters; one series of articles written as decoding \cite{CCL} proven some results; we tried however to follow \cite{LOPCLSET.ABS}, \cite{OPENLATT.ABS}, and \cite{LATTICE4.ABS}. All three were devoted to the Stone representation theorem \cite{StoneRepr} for Boolean or Heyting lattices. The main aim of the present article was to bridge this gap between general distributive lattices and Boolean algebras, having in mind that the more general approach will eventually replace the common proof of aforementioned articles.\footnote{As one of the anonymous reviewers pointed out, it would be interesting to show counterexamples showing that the assumptions of the distributivity and boundedness are necessary, and this will be our plan for future work as basic examples of non-distributive lattices are available as of now only as relational structures.} \par Because in Boolean algebras the notions of ultrafilters, prime filters and maximal filters coincide, we decided to construct some concrete examples of ultrafilters in nontrivial Boolean lattice. We proved also the Prime Ideal Theorem not as BPI (Boolean Prime Ideal), but in the more general setting.\par In the final section we present Nachbin theorems \cite{Nachbin}, \cite{Balbes} expressed both in terms of maximal and prime filters and as the unordered spectra of a lattice \cite{Gratzer2011}, \cite{Gratzer}. This shows that if the notion of maximal and prime filters coincide in the lattice, it is Boolean.

- Eliza Niewiadomska, Adam Grabowski.
{I}ntroduction to Formal Preference Spaces,
Formalized Mathematics 21(3),
pages 223-233, 2013. MML Identifier:
PREFER_1

**Summary**: In the article the formal characterization of preference spaces \cite{Arrow} is given. As the preference relation is one of the very basic notions of mathematical economics \cite{Hallden}, it prepares some ground for a more thorough formalization of consumer theory (although some work has already been done -- see \cite{ARROW.ABS}). There was an attempt to formalize similar results in Mizar, but this work seems still unfinished \cite{Kuzyka}. \par There are many approaches to preferences in literature. We modelled them in a rather illustrative way (similar structures were considered in \cite{PCS_0.ABS}): either the consumer (strictly) prefers an alternative, or they are of equal interest; he/she could also have no opinion of the choice. Then our structures are based on three relations on the (arbitrary, not necessarily finite) set of alternatives. The completeness property can however also be modelled, although we rather follow \cite{Aumann} which is more general \cite{Schumm}. Additionally we assume all three relations are disjoint and their set-theoretic union gives a whole universe of alternatives. \par We constructed some positive and negative examples of preference structures; the main aim of the article however is to give the characterization of consumer preference structures in terms of a binary relation, called characteristic relation \cite{Panek}, and to show the way the corresponding structure can be obtained only using this relation. Finally, we show the connection between tournament and total spaces and usual properties of the ordering relations.

- Noboru Endou, Hiroyuki Okazaki, Yasunari Shidama.
{D}ouble Sequences and Limits,
Formalized Mathematics 21(3),
pages 163-170, 2013. MML Identifier:
DBLSEQ_1
- Number 4
- Maciej Goli\'nski, Artur Korni{\l}owicz.
{C}oproducts in Categories without Uniqueness
of {\bf cod} and {\bf dom},
Formalized Mathematics 21(4),
pages 235-239, 2013. MML Identifier:
ALTCAT_6

**Summary**: The paper introduces coproducts in categories without uniqueness of {\bf cod} and {\bf dom}. It is proven that set-theoretical disjoint union is the coproduct in the category Ens \cite{SEMAD}.

- Mitsuru Jitsukawa, Pauline N. Kawamoto, Yasunari Shidama.
F}ormulation of Cell {P}etri Nets,
Formalized Mathematics 21(4),
pages 241-247, 2013. MML Identifier:
PETRI_3

**Summary**: Based on the Petri net definitions and theorems already formalized in the Mizar article \cite{PETRI.ABS}, in this article we were able to formalize the definition of cell Petri nets. It is based on \cite{Kawamoto-Nakamura:1996}. Colored Petri net has already been defined in \cite{PETRI_2.ABS}. In addition, the conditions of the firing rule and the colored set to this definition, that defines the cell Petri nets are further extended to CPNT.i further. The synthesis of two Petri nets was introduced in \cite{PETRI_2.ABS} and in this work the definition is extended to produce the synthesis of a family of colored Petri nets. Specifically, the extension to a CPNT family is performed by specifying how to link the outbound transitions of each colored Petri net to the place elements of other nets to form a neighborhood relationship. Finally, the activation of colored Petri nets was formalized.

- Yuichi Futa, Noboru Endou, Yasunari Shidama, .
{I}sometric Differentiable Functions on Real Normed Space,
Formalized Mathematics 21(4),
pages 249-260, 2013. MML Identifier:
NDIFF_7

**Summary**: In this article, we formalize isometric differentiable functions on real normed space \cite{Schwartz:1981}, and their properties.

- Keiko Narita, Noboru Endou, Yasunari Shidama.
{D}ifferential Equations on Functions
from {$\mathbb R$} into Real {B}anach Space,
Formalized Mathematics 21(4),
pages 261-272, 2013. MML Identifier:
ORDEQ_02

**Summary**: In this article, we describe the differential equations on functions from $\mathbb R$ into real Banach space. The descriptions are based on the article \cite{ORDEQ_01.ABS}. As preliminary to the proof of these theorems, we proved some properties of differentiable functions on real normed space. For the proof we referred to descriptions and theorems in the article \cite{NDIFF_4.ABS} and the article \cite{NDIFF_5.ABS}. And applying the theorems of Riemann integral introduced in the article \cite{INTEGR21.ABS}, we proved the ordinary differential equations on real Banach space. We referred to the methods of proof in \cite{Schwartz:1967}.

- Yuichi Futa, Hiroyuki Okazaki, Yasunari Shidama.
{S}ubmodule of free {$\mathbb Z$}-module,
Formalized Mathematics 21(4),
pages 273-282, 2013. MML Identifier:
ZMODUL04

**Summary**: In this article, we formalize a free $\mathbb Z$-module and its property. In particular, we formalize the vector space of rational field corresponding to a free $\mathbb Z$-module and prove formally that submodules of a free $\mathbb Z$-module are free. $\mathbb Z$-module is necassary for lattice problems - LLL (Lenstra, Lenstra and Lov\'asz) base reduction algorithm and cryptographic systems with lattice \cite{LATTICE2002}. Some theorems in this article are described by translating theorems in \cite{RLVECT_5.ABS} into theorems of $\mathbb Z$-module, however their proofs are different.

- Maciej Goli\'nski, Artur Korni{\l}owicz.
{C}oproducts in Categories without Uniqueness
of {\bf cod} and {\bf dom},
Formalized Mathematics 21(4),
pages 235-239, 2013. MML Identifier:
ALTCAT_6
**Volume 22, 2014**- Number 1
- Kazuhisa Ishida, Yasunari Shidama, Adam Grabowski.
{D}efinition of Flat Poset and Existence Theorems for Recursive Call,
Formalized Mathematics 22(1),
pages 1-10, 2014. MML Identifier:
POSET_2

**Summary**: This text includes the definition and basic notions of product of posets, chain-complete and flat posets, flattening operation, and the existence theorems of recursive call using the flattening operator. First part of the article, devoted to product and flat posets has a purely mathematical quality. Definition 3 allows to construct a flat poset from arbitrary non-empty set \cite{Davey:2002} in order to provide formal apparatus which eanbles to work with recursive calls within the Mizar langauge. To achieve this we extensively use technical Mizar functors like \verb!BaseFunc! or \verb!RecFunc!. The remaining part builds the background for information engineering approach for lists, namely recursive call for posets \cite{Winskel:1993}. We formalized some facts from Chapter 8 of this book as an introduction to the next two sections where we concentrate on binary product of posets rather than on a more general case.

- Karol P\k{a}k.
{T}ietze Extension Theorem for $n$-dimensional Spaces,
Formalized Mathematics 22(1),
pages 11-19, 2014. MML Identifier:
TIETZE_2

**Summary**: In this article we prove the Tietze extension theorem for an arbitrary convex compact subset of ${\cal E}^n$ with a non-empty interior. This theorem states that, if $T$ is a normal topological space, $X$ is a closed subset of $T$, and $A$ is a convex compact subset of ${\cal E}^n$ with a non-empty interior, then a continuous function $f:X \rightarrow A$ can be extended to a continuous function $g:T\rightarrow {\cal E}^n$. Additionally we show that a subset $A$ is replaceable by an arbitrary subset of a topological space that is homeomorphic with a convex compact subset of ${\cal E}^n$ with a non-empty interior. This article is based on \cite{DUDA:BM61}; \cite{Engelking:1989} and \cite{Engelking:1978} can also serve as reference books.

- Karol P\k{a}k.
{B}rouwer Invariance of Domain Theorem,
Formalized Mathematics 22(1),
pages 21-28, 2014. MML Identifier:
BROUWER3

**Summary**: In this article we focus on a special case of the Brouwer invariance of domain theorem. Let us $A$, $B$ be a subsets of ${\cal E}^n$, and $f:A \rightarrow B$ be a homeomorphic. We prove that, if $A$ is closed then $f$ transform the boundary of $A$ to the boundary of $B$; and if $B$ is closed then $f$ transform the interior of $A$ to the interior of $B$. These two cases are sufficient to prove the topological invariance of dimension, which is used to prove basic properties of the $n$-dimensional manifolds, and also to prove basic properties of the boundary and the interior of manifolds, e.g. the boundary of an $n$-dimension manifold with boundary is an $(n-1)$-dimension manifold. This article is based on \cite{DUDA:BM61}; \cite{Engelking:1989} and \cite{Engelking:1978} can also serve as reference books.

- Pratima K. Shah, Pauline N. Kawamoto, Mariusz Giero.
{T}he Formalization of Decision-Free {P}etri Net,
Formalized Mathematics 22(1),
pages 29-35, 2014. MML Identifier:
PETRI_DF

**Summary**: In this article we formalize the definition of Decision-Free Petri Net (DFPN) presented in \cite{Wang:1998}. Then we formalize the concept of directed path and directed circuit nets in Petri nets to prove properties of DFPN. We also present the definition of firing transitions and transition sequences with natural numbers marking that always check whether transition is enabled or not and after firing it only removes the available tokens (i.e., it does not remove from zero number of tokens). At the end of this article, we show that the total number of tokens in a circuit of decision-free Petri net always remains the same after firing any sequences of the transition.

- Grzegorz Bancerek.
{A}bstract Reduction Systems and Idea of {K}nuth-{B}endix
Completion Algorithm,
Formalized Mathematics 22(1),
pages 37-56, 2014. MML Identifier:
ABSRED_0

**Summary**: Educational content for abstract reduction systems concerning reduction, convertibility, normal forms, divergence and convergence, Church-Rosser property, term rewriting systems, and the idea of the Knuth-Bendix Completion Algorithm. The theory is based on \cite{KlopTRS}.

- Noboru Endou.
{D}ouble Series and Sums,
Formalized Mathematics 22(1),
pages 57-68, 2014. MML Identifier:
DBLSEQ_2

**Summary**: In this paper the author constructs several properties for double series and its convergence. The notions of convergence of double sequence have already been introduced in our previous paper \cite{DBLSEQ_1.ABS}. In section 1 we introduce double series and their convergence. Then we show the relationship between Pringsheim-type convergence and iterated convergence. In section 2 we study double series having non-negative terms. As a result, we have equality of three type sums of non-negative double sequence. In section 3 we show that if a non-negative sequence is summable, then the squence of rearrangement of terms is summable and it has the same sums. In the last section two basic relations between double sequences and matrices are introduced.

- Keiko Narita, Noboru Endou, Yasunari Shidama.
{D}ual Spaces and {H}ahn-{B}anach Theorem,
Formalized Mathematics 22(1),
pages 69-77, 2014. MML Identifier:
DUALSP01

**Summary**: In this article, we deal with dual spaces and the Hahn-Banach Theorem. At the first, we defined dual spaces of real linear spaces and proved related basic properties. Next, we defined dual spaces of real normed spaces. We formed the definitions based on dual spaces of real linear spaces. In addition, we proved properties of the norm about elements of dual spaces. For the proof we referred to descriptions in the article \cite{LOPBAN_1.ABS}. Finally, applying theorems of the second section, we proved the Hahn-Banach extension theorem in real normed spaces. We have used extensively used \cite{HAHNBAN.ABS}.

- Roland Coghetto.
{S}emiring of Sets,
Formalized Mathematics 22(1),
pages 79-84, 2014. MML Identifier:
SRINGS_1

**Summary**: Schmets \cite{Schmets:2004} has developed a measure theory from a generalized notion of a semiring of sets. Goguadze \cite{GOGUADZE:2003} has introduced another generalized notion of semiring of sets and proved that all known properties that semiring have according to the old definitions are preserved. We show that this two notions are almost equivalent. We note that Patriota \cite{2011arXiv1103.6166P} has defined this quasi-semiring. We propose the formalization of some properties developed by the authors.

- Roland Coghetto.
{S}emiring of Sets: Examples,
Formalized Mathematics 22(1),
pages 85-88, 2014. MML Identifier:
SRINGS_2

**Summary**: This article proposes the formalization of some examples of semiring of sets proposed by Goguadze \cite{GOGUADZE:2003} and Schmets \cite{Schmets:2004}.

- Adam Grabowski.
{T}opological Interpretation of Rough Sets,
Formalized Mathematics 22(1),
pages 89-97, 2014. MML Identifier:
ROUGHS_4

**Summary**: Rough sets, developed by Pawlak, are an important model of incomplete or partially known information. In this article, which is essentially a continuation of \cite{ROUGHS_1.ABS}, we characterize rough sets in terms of topological closure and interior, as the approximations have the properties of the Kuratowski operators. We decided to merge topological spaces with tolerance approximation spaces. As a testbed for our developed approach, we restated the results of Isomichi \cite{ISOMICHI} (formalized in Mizar in \cite{ISOMICHI.ABS}) and about fourteen sets of Kuratowski \cite{KURAT:4} (encoded with the help of Mizar adjectives and clusters' registrations in \cite{KURATO_1.ABS}) in terms of rough approximations. The upper bounds which were 14 and 7 in the original paper of Kuratowski, in our case are six and three, respectively. \par It turns out that within the classification given by Isomichi, $1^{\rm st}$ class subsets are precisely crisp sets, $2^{\rm nd}$ class subsets are proper rough sets, and there are no $3^{\rm rd}$ class subsets in topological spaces generated by approximations. Also the important results about these spaces is that they are extremally disconnected \cite{TDLAT_3.ABS}, hence lattices of their domains are Boolean. \par Furthermore, we develop the theory of abstract spaces equipped with maps possessing characteristic properties of rough approximations which enables us to freely use the notions from the theory of rough sets and topological spaces formalized in the Mizar Mathematical Library \cite{GrabowskiFI:2013}.

- Kazuhisa Ishida, Yasunari Shidama, Adam Grabowski.
{D}efinition of Flat Poset and Existence Theorems for Recursive Call,
Formalized Mathematics 22(1),
pages 1-10, 2014. MML Identifier:
POSET_2
- Number 2
**Special Issue: 25 years of the Mizar Mathematical Library**

- Adam Grabowski and Yasunari Shidama.
Preface,
Formalized Mathematics 22(2),
pages i-iv, 2014.

- Marco B. Caminati and Artur Korni{\l}owicz.
{P}seudo-Canonical Formulae are Classical,
Formalized Mathematics 22(2),
pages 99-103, 2014. MML Identifier:
HILBERT4

**Summary**: An original result about Hilbert Positive Propositional Calculus introduced in \cite{HILBERT1.ABS} is proven. That is, it is shown that the pseudo-canonical formulae of that calculus (and hence also the canonical ones, see \cite{HILBERT3.ABS}) are a subset of the classical tautologies.

- Yasushige Watase.
{L}agrange's Four-Square Theorem,
Formalized Mathematics 22(2),
pages 105-110, 2014. MML Identifier:
LAGRA4SQ

**Summary**: This article provides a formalized proof of the so-called ``the four-square theorem", namely any natural number can be expressed by a sum of four squares, which was proved by Lagrange in 1770. An informal proof of the theorem can be found in the number theory literature, e.g. in \cite{HardyWright}, \cite{Baker:1984} or \cite{Wada:1981}. \par This theorem is item {\tt{\#19}} from the ``Formalizing 100 Theorems" list maintained by Freek Wiedijk at \url{http://www.cs.ru.nl/F.Wiedijk/100/} .

- Christoph Schwarzweller.
{P}roth Numbers,
Formalized Mathematics 22(2),
pages 111-118, 2014. MML Identifier:
NAT_6

**Summary**: In this article we introduce Proth numbers and prove two theorems on such numbers being prime \cite{Buchmann:1992}. We also give revised versions of Pocklington's theorem and of the Legendre symbol. Finally, we prove Pepin's theorem and that the fifth Fermat number is not prime.

- Karol P\k{a}k.
{B}ertrand's Ballot Theorem,
Formalized Mathematics 22(2),
pages 119-123, 2014. MML Identifier:
BALLOT_1

**Summary**: In this article we formalize the Bertrand's Ballot Theorem based on \cite{BALLOT}. Suppose that in an election we have two candidates: $A$ that receives $n$ votes and $B$ that receives $k$ votes, and additionally $n\geq k$. Then this theorem states that the probability of the situation where $A$ maintains more votes than $B$ throughout the counting of the ballots is equal to $(n-k)/(n+k)$. \par This theorem is item {\tt{\#30}} from the ``Formalizing 100 Theorems" list maintained by Freek Wiedijk at \url{http://www.cs.ru.nl/F.Wiedijk/100/} .

- Grzegorz Bancerek.
{T}erm Context,
Formalized Mathematics 22(2),
pages 125-155, 2014. MML Identifier:
MSAFREE5

**Summary**: Two construction functors: simple term with a variable and compound term with an operation and argument terms and schemes of term induction are introduced. The degree of construction as a number of used operation symbols is defined. Next, the term context is investigated. An $x$-context is a term which includes a variable $x$ once only. The compound term is $x$-context iff the argument terms include an $x$-context once only. The context induction is shown and used many times. As a key concept, the context substitution is introduced. Finally, the translations and endomorphisms are expressed by context substitution.

- Adam Grabowski.
{C}auchy Mean Theorem,
Formalized Mathematics 22(2),
pages 157-166, 2014. MML Identifier:
RVSUM_3

**Summary**: The purpose of this paper was to prove formally, using the Mizar language, Arithmetic Mean/Geometric Mean theorem known maybe better under the name of AM-GM inequality or Cauchy mean theorem. It states that the arithmetic mean of a list of a non-negative real numbers is greater than or equal to the geometric mean of the same list. \par The formalization was tempting for at least two reasons: one of them, perhaps the strongest, was that the proof of this theorem seemed to be relatively easy to formalize (e.g. the weaker variant of this was proven in \cite{SERIES_3.ABS}). Also Jensen's inequality is already present in the Mizar Mathematical Library. We were impressed by the beauty and elegance of the simple proof by induction and so we decided to follow this specific way. \par The proof follows similar lines as that written in Isabelle \cite{Cauchy-AFP}; the comparison of both could be really interesting as it seems that in both systems the number of lines needed to prove this are really close. \par This theorem is item {\tt{\#38}} from the ``Formalizing 100 Theorems" list maintained by Freek Wiedijk at \url{http://www.cs.ru.nl/F.Wiedijk/100/} .

- William Richter, Adam Grabowski and Jesse Alama.
{T}arski Geometry Axioms,
Formalized Mathematics 22(2),
pages 167-176, 2014. MML Identifier:
GTARSKI1

**Summary**: This is the translation of the Mizar article containing readable Mizar proofs of some axiomatic geometry theorems formulated by the great Polish mathematician Alfred Tarski \cite{TarskiGivant}, and we hope to continue this work.\footnote{The first author ported the code to HOL Light (\url{http://www.cl.cam.ac.uk/~jrh13/hol-light/}), which can be found in any recent subversion of HOL Light as {\tt hol\_light/RichterHilbertAxiomGeometry/TarskiAxiomGeometry\_read.ml}} \par The article is an extension and upgrading of the source code written by the first author with the help of \verb!miz3! tool; his primary goal was to use proof checkers to help teach rigorous axiomatic geometry in high school using Hilbert's axioms. \par This is largely a Mizar port of Julien Narboux's Coq pseudo-code \cite{Narboux:2007}. We partially prove the theorem of \cite{Schwabhauser:1983} that Tarski's (extremely weak!) plane geometry axioms imply Hilbert's axioms. Specifically, we obtain Gupta's amazing proof which implies Hilbert's axiom \verb!I1! that two points determine a line. \par The primary Mizar coding was heavily influenced by \cite{INCSP_1.ABS} on axioms of incidence geometry. The original development was much improved using Mizar adjectives instead of predicates only, and to use this machinery in full extent, we have to construct some models of Tarski geometry. These are listed in the second section, together with appropriate registrations of clusters. Also models of Tarski's geometry related to real planes were constructed.

- Adam Naumowicz.
{A} Note on the Seven Bridges of {K}\"onigsberg Problem,
Formalized Mathematics 22(2),
pages 177-178, 2014. MML Identifier:
GRAPH_3A

**Summary**: In this paper we account for the formalization of the seven bridges of K\"onigsberg puzzle. The problem originally posed and solved by Euler in 1735 is historically notable for having laid the foundations of graph theory, cf. \cite{Chartrand:1985}. Our formalization utilizes a simple set-theoretical graph representation with four distinct sets for the graph's vertices and another seven sets that represent the edges (bridges). The work appends the article by Nakamura and Rudnicki \cite{GRAPH_3.ABS} by introducing the classic example of a graph that does not contain an Eulerian path. \par This theorem is item {\tt{\#54}} from the ``Formalizing 100 Theorems" list maintained by Freek Wiedijk at \url{http://www.cs.ru.nl/F.Wiedijk/100/} .

- Karol P\k{a}k.
{T}opological Manifolds,
Formalized Mathematics 22(2),
pages 179-186, 2014. MML Identifier:
MFOLD_0

**Summary**: Let us recall that a topological space $M$ is a topological manifold if $M$ is second-countable Hausdorff and locally Euclidean, i.e. each point has a neighborhood that is homeomorphic to an open ball of ${\cal E}^n$ for some $n$. However, if we would like to consider a topological manifold with a boundary, we have to extend this definition. Therefore, we introduce here the concept of a locally Euclidean space that covers both cases (with and without a boundary), i.e. where each point has a neighborhood that is homeomorphic to a closed ball of ${\cal E}^n$ for some $n$. \par Our purpose is to prove, using the Mizar formalism, a number of properties of such locally Euclidean spaces and use them to demonstrate basic properties of a manifold. Let $T$ be a locally Euclidean space. We prove that every interior point of $T$ has a neighborhood homeomorphic to an open ball and that every boundary point of $T$ has a neighborhood homeomorphic to a closed ball, where additionally this point is transformed into a point of the boundary of this ball. When $T$ is $n$-dimensional, i.e. each point of $T$ has a neighborhood that is homeomorphic to a closed ball of ${\cal E}^n$, we show that the interior of $T$ is a locally Euclidean space without boundary of dimension $n$ and the boundary of $T$ is a locally Euclidean space without boundary of dimension $n-1$. Additionally, we show that every connected component of a compact locally Euclidean space is a locally Euclidean space of some dimension. We prove also that the Cartesian product of locally Euclidean spaces also forms a locally Euclidean space. We determine the interior and boundary of this product and show that its dimension is the sum of the dimensions of its factors. At the end, we present several consequences of these results for topological manifolds. This article is based on \cite{ENGEL:BM51}.

- Adam Grabowski and Yasunari Shidama.
Preface,
Formalized Mathematics 22(2),
pages i-iv, 2014.
- Number 3

- Kazuhisa Nakasho, Yuichi Futa, Hiroyuki Okazaki and Yasunari Shidama,.
{R}ank of Submodule, Linear
Transformations and Linearly Independent Subsets of {$\mathbb{Z}$}-module,
Formalized Mathematics 22(3),
pages 189-198, 2014. MML Identifier:
ZMODUL05

**Summary**: In this article, we formalize some basic facts of $\mathbb Z$-module. In the first section, we discuss the rank of submodule of $\mathbb Z$-module and its properties. Especially, we formally prove that the rank of any $\mathbb Z$-module is equal to or more than that of its submodules, and vice versa, and that there exists a submodule with any given rank that satisfies the above condition. In the next section, we mention basic facts of linear transformations between two $\mathbb Z$-modules. In this section, we define homomorphism between two $\mathbb Z$-modules and deal with kernel and image of homomorphism. In the last section, we formally prove some basic facts about linearly independent subsets and linear combinations. These formalizations are based on \cite{BourbakiAlgI}(p.191-242), \cite{lang-algebra}(p.117-172) and \cite{atiyah1969introduction}(p.17-35).

- Peter Jaeger.
{E}vents of {B}orel Sets, Construction of {B}orel
Sets and Random Variables for Stochastic Finance,
Formalized Mathematics 22(3),
pages 199-204, 2014. MML Identifier:
FINANCE2

**Summary**: We consider special events of Borel sets with the aim to prove, that the set of the irrational numbers is an event of the Borel sets. The set of the natural numbers, the set of the integer numbers and the set of the rational numbers are countable, so we can use the literature \cite{analysis1:2001} (pp. 78-81) as a basis for the similar construction of the proof. Next we prove, that different sets can construct the Borel sets \cite{klenke:2006} (pp. 9-10). Literature \cite{klenke:2006} (pp. 9-10) and \cite{georgii:2004} (pp. 11-12) gives an overview, that there exists some other sets for this construction. Last we define special functions as random variables for stochastic finance in discrete time. The relevant functions are implemented in the article \cite{FINANCE1.ABS}, see \cite{follmerschied:2004} (p. 4). The aim is to construct events and random variables, which can easily be used with a probability measure. See as an example theorems (10) and (14) in \cite{RANDOM_3.ABS}. Then the formalization is more similar to the presentation used in the book \cite{follmerschied:2004}. As a background, further literatures is \cite{bosch:2008} (pp. 9-12), \cite{analysisheuser:2003} (pp. 17-20), and \cite{linalgfischer:2002} (pp.32-35).

- Rafa{\l} Ziobro.
{S}ome Remarkable Identities Involving Numbers,
Formalized Mathematics 22(3),
pages 205-208, 2014. MML Identifier:
NEWTON01

**Summary**: The article focuses on simple identities found for binomials, their divisibility, and basic inequalities. A general formula allowing factorization of the sum of like powers is introduced and used to prove elementary theorems for natural numbers. \par Formulas for short multiplication are sometimes referred in English or French as remarkable identities. The same formulas could be found in works concerning polynomial factorization, where there exists no single term for various identities. Their usability is not questionable, and they have been successfully utilized since for ages. For example, in his books published in 1731 (p. 385), Edward Hatton \cite{Hatton1731} wrote: ``Note, that the differences of any two like powers of two quantities, will always be divided by the difference of the quantities without any remainer...". \par Despite of its conceptual simplicity, the problem of factorization of sums/diffe\-ren\-ces of two like powers could still be analyzed \cite{Nowak1998}, giving new and possibly interesting results \cite{Mostafa2005}.

- Kazuhisa Nakasho, Yuichi Futa and Yasunari Shidama.
{T}opological Properties of Real Normed Space,
Formalized Mathematics 22(3),
pages 209-223, 2014. MML Identifier:
NORMSP_3

**Summary**: In this article, we formalize topological properties of real normed spaces. In the first part, open and closed, density, separability and sequence and its convergence are discussed. Then we argue properties of real normed subspace. Then we discuss linear functions between real normed speces. Several kinds of subspaces induced by linear functions such as kernel, image and inverse image are considered here. The fact that Lipschitz continuity operators preserve convergence of sequences is also refered here. Then we argue the condition when real normed subspaces become Banach's spaces. We also formalize quotient vector space. In the last session, we argue the properties of the closure of real normed space. These formalizations are based on \cite{rudin1991functional}(p.3-41), \cite{bourbaki1987elements} and \cite{yoshida:1980}(p.3-67).

- Grzegorz Bancerek.
{A}lgebraic Approach to Algorithmic Logic,
Formalized Mathematics 22(3),
pages 225-255, 2014. MML Identifier:
AOFA_L00

**Summary**: We introduce algorithmic logic -- an algebraic approach according to \cite{Salwicki}. It is done in three stages: propositional calculus, quantifier calculus with equality, and finally proper algorithmic logic. For each stage appropriate signature and theory are defined. Propositional calculus and quantifier calculus with equality are explored according to \cite{Mendelson}. A language is introduced with language signature including free variables, substitution, and equality. Algorithmic logic requires a bialgebra structure which is an extension of language signature and program algebra. While-if algebra of generator set and algebraic signature is bialgebra with appropriate properties and is used as basic type of algebraic logic.

- Adam Grabowski.
{F}ormalization of Generalized Almost Distributive Lattices,
Formalized Mathematics 22(3),
pages 257-267, 2014. MML Identifier:
LATTAD_1

**Summary**: Almost Distributive Lattices (ADL) are structures defined by Swamy and Rao \cite{SwamyRao1981} as a common abstraction of some generalizations of the Boolean algebra. In our paper, we deal with a certain further generalization of ADLs, namely the Generalized Almost Distributive Lattices (GADL). Our main aim was to give the formal counterpart of this structure and we succeeded formalizing all items from the Section 3 of {R}ao et al.'s paper \cite{GADL:2009}. Essentially among GADLs we can find structures which are neither $\vee$-commutative nor $\wedge$-commutative (resp., $\wedge$-commutative); consequently not all forms of absorption identities hold.\par We characterized some necessary and sufficient conditions for commutativity and distributivity, we also defined the class of GADLs with zero element. We tried to use as much attributes and cluster registrations as possible, hence many identities are expressed in terms of adjectives; also some generalizations of well-known notions from lattice theory \cite{Gratzer2011} formalized within the Mizar Mathematical Library were proposed. Finally, some important examples from Rao's paper were introduced. We construct the example of GADL which is not an ADL. Mechanization of proofs in this specific area could be a good starting point towards further generalization of lattice theory \cite{Gratzer} with the help of automated theorem provers \cite{GrabMo2004}.

- Kenichi Arai, Ken Wakabayashi and Hiroyuki Okazaki.
{D}ifference of Function
on~Vector~Space~over~\hbox{$\mathbb F$},
Formalized Mathematics 22(3),
pages 269-275, 2014. MML Identifier:
VSDIFF_1

**Summary**: In \cite{DIFF_1.ABS}, the definitions of forward difference, backward difference, and central difference as difference operations for functions on $\mathbb{R}$ were formalized. However, the definitions of forward difference, backward difference, and central difference for functions on vector spaces over $\mathbb{F}$ have not been formalized. In cryptology, these definitions are very important in evaluating the security of cryptographic systems \cite{BA1991}, \cite{Lai1994}. Differential cryptanalysis \cite{BA1993} that undertakes a general purpose attack against block ciphers \cite{DESCIP_1.ABS} can be formalized using these definitions. In this article, we formalize the definitions of forward difference, backward difference, and central difference for functions on vector spaces over $\mathbb{F}$. Moreover, we formalize some facts about these definitions.

- Kazuhisa Nakasho, Yuichi Futa, Hiroyuki Okazaki and Yasunari Shidama,.
{R}ank of Submodule, Linear
Transformations and Linearly Independent Subsets of {$\mathbb{Z}$}-module,
Formalized Mathematics 22(3),
pages 189-198, 2014. MML Identifier:
ZMODUL05
- Number 4

- Yuichi Futa, Hiroyuki Okazaki, Kazuhisa Nakasho and Yasunari Shidama.
{T}orsion {$\mathbb{Z}$}-module and
Torsion-free {$\mathbb{Z}$}-module,
Formalized Mathematics 22(4),
pages 277-289, 2014. MML Identifier:
ZMODUL06

**Summary**: In this article, we formalize a torsion $\mathbb Z$-module and a torsion-free $\mathbb Z$-module. Especially, we prove formally that finitely generated torsion-free $\mathbb Z$-modules are finite rank free. We also formalize properties related to rank of finite rank free $\mathbb Z$-modules. The notion of $\mathbb Z$-module is necessary for solving lattice problems, LLL (Lenstra, Lenstra, and Lov\'asz) base reduction algorithm \cite{LLL}, cryptographic systems with lattice \cite{LATTICE2002}, and coding theory \cite{LANDC}.

- Artur Korni{\l}owicz and Christoph Schwarzweller.
{T}he First Isomorphism Theorem and Other Properties of Rings,
Formalized Mathematics 22(4),
pages 291-301, 2014. MML Identifier:
RING_2

**Summary**: Different properties of rings and fields are discussed \cite{Jacobson2009}, \cite{Waerden2003} and \cite{Luneburg1999}. We introduce ring homomorphisms, their kernels and images, and prove the First Isomorphism Theorem, namely that for a homomorphism $f: R\longrightarrow S$ we have $R/_{\mbox{ker}(f)} \cong {\mbox{Im}(f)}$. Then we define prime and irreducible elements and show that every principal ideal domain is factorial. Finally we show that polynomial rings over fields are Euclidean and hence also factorial.

- Keiko Narita, Noboru Endou and Yasunari Shidama.
{B}idual Spaces and Reflexivity of Real Normed Spaces,
Formalized Mathematics 22(4),
pages 303-311, 2014. MML Identifier:
DUALSP02

**Summary**: In this article, we considered bidual spaces and reflexivity of real normed spaces. At first we proved some corollaries applying Hahn-Banach theorem and showed related theorems. In the second section, we proved the norm of dual spaces and defined the natural mapping, from real normed spaces to bidual spaces. We also proved some properties of this mapping. Next, we defined real normed space of {$\mathbb R$}, real number spaces as real normed spaces and proved related theorems. We can regard linear functionals as linear operators by this definition. Accordingly we proved Uniform Boundedness Theorem for linear functionals using the theorem (5) from \cite{LOPBAN_5.ABS}. Finally, we defined reflexivity of real normed spaces and proved some theorems about isomorphism of linear operators. Using them, we proved some properties about reflexivity. These formalizations are based on \cite{ReedSimon1972}, \cite{rudin1991functional}, \cite{Dax2002} and \cite{Brezis2011}.

- Roland Coghetto.
{S}ome Facts about Trigonometry and {E}uclidean Geometry,
Formalized Mathematics 22(4),
pages 313-319, 2014. MML Identifier:
EUCLID10

**Summary**: We calculate the values of the trigonometric functions for angles: $\frac{\pi}{3}$ and $\frac{\pi}{6}$, by \cite{hartshorne2000geometry}. After defining some trigonometric identities, we demonstrate conventional trigonometric formulas in the triangle, and the geometric property, by \cite{efimov1981geometrie}, of the triangle inscribed in a semicircle, by the proposition 3.31 in \cite{fitzpatrick2007euclid}. Then we define the diameter of the circumscribed circle of a triangle using the definition of the area of a triangle and prove some identities of a triangle \cite{Coxeter:1967}. We conclude by indicating that the diameter of a circle is twice the length of the radius.

- Adam Grabowski.
{T}he Formal Construction of Fuzzy Numbers,
Formalized Mathematics 22(4),
pages 321-327, 2014. MML Identifier:
FUZNUM_1

**Summary**: In this article, we continue the development of the theory of fuzzy sets \cite{Zadeh:1965}, started with \cite{FUZZY_1.ABS} with the future aim to provide the formalization of fuzzy numbers \cite{DuboisPrade:1978} in terms reflecting the current state of the Mizar Mathematical Library. Note that in order to have more usable approach in \cite{FUZZY_1.ABS}, we revised that article as well; some of the ideas were described in \cite{GrabowskiFuzzy:2013}. As we can actually understand fuzzy sets just as their membership functions (via the equality of membership function and their set-theoretic counterpart), all the calculations are much simpler. To test our newly proposed approach, we give the notions of (normal) triangular and trapezoidal fuzzy sets as the examples of concrete fuzzy objects. Also $\alpha$-cuts, the core of a fuzzy set, and normalized fuzzy sets were defined. Main technical obstacle was to prove continuity of the glued maps, and in fact we did this not through its topological counterpart, but extensively reusing properties of the real line (with loss of generality of the approach, though), because we aim at formalizing fuzzy numbers in our future submissions, as well as merging with rough set approach as introduced in \cite{ROUGHS_1.ABS} and \cite{GrabowskiFI:2014}. Our base for formalization was \cite{DuboisPrade:1980} and \cite{DuboisPrade:1990}.

- Yuichi Futa, Hiroyuki Okazaki, Kazuhisa Nakasho and Yasunari Shidama.
{T}orsion {$\mathbb{Z}$}-module and
Torsion-free {$\mathbb{Z}$}-module,
Formalized Mathematics 22(4),
pages 277-289, 2014. MML Identifier:
ZMODUL06
**Volume 23, 2015**- Number 1

- Marco Riccardi.
{C}ategorical Pullbacks,
Formalized Mathematics 23(1),
pages 1-14, 2015. MML Identifier:
CAT_7

**Summary**: The main purpose of this article is to introduce the categorical concept of pullback in Mizar. In the first part of this article we redefine hom-sets, monomorphisms, epimorpshisms and isomorphisms \cite{Borceaux} within a free-object category \cite{Adamek:2009} and it is shown there that ordinal numbers can be considered as categories. Then the pullback is introduced in terms of its universal property and the Pullback Lemma is formalized \cite{MacLane:1}. In the last part of the article we formalize the pullback of functors \cite{Lawvere:1963} and it is also shown that it is not possible to write an equivalent definition in the context of the previous Mizar formalization of category theory \cite{CAT_1.ABS}.

- Kazuhisa Nakasho, Hiroshi Yamazaki, Hiroyuki Okazaki and Yasunari Shidama.
{D}efinition and Properties of Direct Sum Decomposition of Groups,
Formalized Mathematics 23(1),
pages 15-27, 2015. MML Identifier:
GROUP_19

**Summary**: In this article, direct sum decomposition of group is mainly discussed. In the second section, support of element of direct product group is defined and its properties are formalized. It is formalized here that an element of direct product group belongs to its direct sum if and only if support of the element is finite. In the third section, product map and sum map are prepared. In the fourth section, internal and external direct sum are defined. In the last section, an equivalent form of internal direct sum is proved. We referred to \cite{rotman1995introduction}, \cite{robinson2012course}, \cite{BourbakiAlgI} and \cite{lang-algebra} in the formalization.

- Yuichi Futa, Hiroyuki Okazaki and Yasunari Shidama.
{M}atrix of {$\mathbb Z$}-module,
Formalized Mathematics 23(1),
pages 29-49, 2015. MML Identifier:
ZMATRLIN

**Summary**: In this article, we formalize a matrix of $\mathbb Z$-module and its properties. Specially, we formalize a matrix of a linear transformation of $\mathbb Z$-module, a bilinear form and a matrix of the bilinear form (Gramian matrix). We formally prove that for a finite-rank free $\mathbb Z$-module $V$, determinant of its Gramian matrix is constant regardless of selection of its basis. $\mathbb Z$-module is necessary for lattice problems, LLL (Lenstra, Lenstra and Lov\'asz) base reduction algorithm and cryptographic systems with lattices \cite{LATTICE2002} and coding theory \cite{LANDC}. Some theorems in this article are described by translating theorems in \cite{MATRLIN.ABS}, \cite{HAHNBAN.ABS} and \cite{BILINEAR.ABS} into theorems of $\mathbb Z$-module.

- Noboru Endou, Kazuhisa Nakasho and Yasunari Shidama.
{$\sigma$}-ring and {$\sigma$}-algebra of Sets,
Formalized Mathematics 23(1),
pages 51-57, 2015. MML Identifier:
SRINGS_3

**Summary**: In this article, semiring and semialgebra of sets are formalized so as to construct a measure of a given set in the next step. Although a semiring of sets has already been formalized in \cite{SRINGS_1.ABS}, that is, strictly speaking, a definition of a quasi semiring of sets suggested in the last few decades \cite{GOGUADZE:2003}. We adopt a classical definition of a semiring of sets here to avoid such a confusion. Ring of sets and algebra of sets have been formalized as non empty preboolean set \cite{FINSUB_1.ABS} and field of subsets \cite{PROB_1.ABS}, respectively. In the second section, definitions of a ring and a $\sigma$-ring of sets, which are based on a semiring and a ring of sets respectively, are formalized and their related theorems are proved. In the third section, definitions of an algebra and a $\sigma$-algebra of sets, which are based on a semialgebra and an algebra of sets respectively, are formalized and their related theorems are proved. In the last section, mutual relationships between $\sigma$-ring and $\sigma$-algebra of sets are formalized and some related examples are given. The formalization is based on \cite{GOGUADZE:2003}, and also referred to \cite{bogachev2007measure} and \cite{Halmos74}.

- Kazuhisa Nakasho and Noboru Endou.
{S}eparability of Real Normed Spaces and Its Basic Properties,
Formalized Mathematics 23(1),
pages 59-65, 2015. MML Identifier:
NORMSP_4

**Summary**: In this article, the separability of real normed spaces and its properties are mainly formalized. In the first section, it is proved that a real normed subspace is separable if it is generated by a countable subset. We used here the fact that the rational numbers form a dense subset of the real numbers. In the second section, the basic properties of the separable normed spaces are discussed. It is applied to isomorphic spaces via bounded linear operators and double dual spaces. In the last section, it is proved that the completeness and reflexivity are transferred to sublinear normed spaces. The formalization is based on \cite{yoshida:1980}, and also referred to \cite{BOURBAKI:1-5}, \cite{Dunford:1958} and \cite{kolmogorov2012}.

- Kazuhisa Nakasho, Hiroyuki Okazaki, Hiroshi Yamazaki and Yasunari Shidama.
{E}quivalent Expressions of Direct Sum Decomposition of Groups,
Formalized Mathematics 23(1),
pages 67-73, 2015. MML Identifier:
GROUP_20

**Summary**: In this article, the equivalent expressions of the direct sum decomposition of groups are mainly discussed. In the first section, we formalize the fact that the internal direct sum decomposition can be defined as normal subgroups and some of their properties. In the second section, we formalize an equivalent form of internal direct sum of commutative groups. In the last section, we formalize that the external direct sum leads an internal direct sum. We referred to \cite{rotman1995introduction}, \cite{robinson2012course} \cite{BourbakiAlgI} and \cite{lang-algebra} in the formalization.

- Marco Riccardi.
{C}ategorical Pullbacks,
Formalized Mathematics 23(1),
pages 1-14, 2015. MML Identifier:
CAT_7
- Number 2

- Roland Coghetto.
{M}orley's Trisector Theorem,
Formalized Mathematics 23(2),
pages 75-79, 2015. MML Identifier:
EUCLID11

**Summary**: Morley's trisector theorem states that ``The points of intersection of the adjacent trisectors of the angles of any triangle are the vertices of an equilateral triangle" \cite{Coxeter:1967}. \par There are many proofs of Morley's trisector theorem \cite{donolato2013vector, maor2014beautiful, MAI1, MAI2, connes1998new, stonebridge2009simple, CTK, oakley1978morley}. We follow the proof given by A. Letac in \cite{letac}.

- Karol P\k{a}k.
{F}lexary Operations,
Formalized Mathematics 23(2),
pages 81-92, 2015. MML Identifier:
FLEXARY1

**Summary**: In this article we introduce necessary notation and definitions to prove the Euler's Partition Theorem according to H.S. Wilf's lecture notes \cite{Wilf}. Our aim is to create an environment which allows to formalize the theorem in a way that is as similar as possible to the original informal proof. \par {E}uler's {P}artition {T}heorem is listed as item {\tt{\#45}} from the ``Formalizing 100 Theorems" list maintained by Freek Wiedijk at \url{http://www.cs.ru.nl/F.Wiedijk/100/} \cite{Freek-100-theorems}.

- Karol P\k{a}k.
{E}uler's {P}artition {T}heorem,
Formalized Mathematics 23(2),
pages 93-99, 2015. MML Identifier:
EULRPART

**Summary**: In this article we prove the Euler's Partition Theorem which states that the number of integer partitions with odd parts equals the number of partitions with distinct parts. The formalization follows H.S. Wilf's lecture notes \cite{Wilf} (see also \cite{Andrews}). \par {E}uler's {P}artition {T}heorem is listed as item {\tt{\#45}} from the ``Formalizing 100 Theorems" list maintained by Freek Wiedijk at \url{http://www.cs.ru.nl/F.Wiedijk/100/} \cite{Freek-100-theorems}.

- Yasushige Watase.
{I}ntroduction to {D}iophantine Approximation,
Formalized Mathematics 23(2),
pages 101-106, 2015. MML Identifier:
DIOPHAN1

**Summary**: In this article we formalize some results of Diophantine approximation, i.e. the approximation of an irrational number by rationals. A typical example is finding an integer solution $(x,y)$ of the inequality $| x\theta - y | \leq 1/x$, where $\theta$ is a real number. First, we formalize some lemmas about continued fractions. Then we prove that the inequality has infinitely many solutions by continued fractions. Finally, we formalize Dirichlet's proof (1842) of existence of the solution \cite{HardyWright}, \cite{Baker:1984}.

- Roland Coghetto.
{F}inite Product of Semiring of Sets,
Formalized Mathematics 23(2),
pages 107-114, 2015. MML Identifier:
SRINGS_4

**Summary**: We formalize that the image of a semiring of sets \cite{GOGUADZE:2003} by an injective function is a semiring of sets. We offer a non-trivial example of a semiring of sets in a topological space \cite{Schmets:2004}. Finally, we show that the finite product of a semiring of sets is also a semiring of sets \cite{Schmets:2004} and that the finite product of a classical semiring of sets \cite{bogachev2007measure} is a classical semiring of sets. In this case, we use here the notation from the book of Aliprantis and Border \cite{aliprantis2006infinite}.

- Adam Grabowski.
{T}wo Axiomatizations of {N}elson Algebras,
Formalized Mathematics 23(2),
pages 115-125, 2015. MML Identifier:
NELSON_1

**Summary**: Nelson algebras were first studied by Rasiowa and Bia\l ynicki-Birula \cite{RasiowaBirula} under the name N-lattices or quasi-pseudo-Boolean algebras. Later, in investigations by Monteiro and Brignole \cite{BrignoleI:1967, BrignoleII:1967}, and \cite{Brignole} the name ``Nelson algebras" was adopted -- which is now commonly used to show the correspondence with Nelson's paper \cite{Nelson} on constructive logic with strong negation.\par By a Nelson algebra we mean an abstract algebra $$\langle L, \top, -, \neg, \rightarrow, \Rightarrow, \sqcup, \sqcap \rangle$$ \noindent where $L$ is the carrier, $-$ is a quasi-complementation (Rasiowa used the sign $\sim$, but in Mizar ``$-$" should be used to follow the approach described in \cite{GrabMo2004} and \cite{GrabowskiJAR40}), $\neg$ is a weak pseudo-complementation, $\rightarrow$ is weak relative pseudo-complementation and $\Rightarrow$ is implicative operation. $\sqcup$ and $\sqcap$ are ordinary lattice binary operations of supremum and infimum. \par In this article we give the definition and basic properties of these algebras according to \cite{RasiowaNonClassical} and \cite{Rasiowa:2001}. We start with preliminary section on quasi-Boolean algebras (i.e. de Morgan bounded lattices). Later we give the axioms in the form of Mizar adjectives with names corresponding with those in \cite{Rasiowa:2001}. As our main result we give two axiomatizations (non-equational and equational) and the full formal proof of their equivalence. \par The second set of equations is rather long but it shows the logical essence of Nelson lattices. This formalization aims at the construction of algebraic model of rough sets \cite{GrabowskiFI:2013} in our future submissions. Section 4 contains all items from Th. 1.2 and 1.3 (and the itemization is given in the text). In the fifth section we provide full formal proof of Th. 2.1 p. 75 \cite{RasiowaNonClassical}.

- Roland Coghetto.
{G}roups -- Additive Notation,
Formalized Mathematics 23(2),
pages 127-160, 2015. MML Identifier:
GROUP_1A

**Summary**: We translate the articles covering group theory already available in the Mizar Mathematical Library from multiplicative into additive notation. We adapt the works of Wojciech A. Trybulec \cite{GROUP_1.ABS, GROUP_2.ABS, GROUP_3.ABS} and Artur Korni{\l}owicz \cite{TOPGRP_1.ABS}. \par In particular, these authors have defined the notions of group, abelian group, power of an element of a group, order of a group and order of an element, subgroup, coset of a subgroup, index of a subgroup, conjugation, normal subgroup, topological group, dense subset and basis of a topological group. Lagrange's theorem and some other theorems concerning these notions \cite{blahut2014cryptography, inui2012group, hewitt2012abstract} are presented. \par Note that ``The term {$\mathbb Z$}-module is simply another name for an additive abelian group" \cite{NC2012}. We take an approach different than that used by Futa et al. \cite{ZMODUL01.ABS} to use in a future article the results obtained by Artur Korni{\l}owicz \cite{TOPGRP_1.ABS}. Indeed, H{\"o}lzl et al. showed that it was possible to build ``a generic theory of limits based on filters" in Isabelle/HOL \cite{holzl2013type, boldo2014formalization}. Our goal is to define the convergence of a sequence and the convergence of a series in an abelian topological group \cite{bourbaki2013general} using the notion of filters.

- Roland Coghetto.
{M}orley's Trisector Theorem,
Formalized Mathematics 23(2),
pages 75-79, 2015. MML Identifier:
EUCLID11
- Number 3

- Taneli Huuskonen.
{P}olish Notation,
Formalized Mathematics 23(3),
pages 161-176, 2015. MML Identifier:
POLNOT_1

**Summary**: This article is the first in a series formalizing some results in my joint work with Prof.\ Joanna Goli\'nska-Pilarek (\cite{GPH:2012} and \cite{GPH:2015}) concerning a logic proposed by Prof. Andrzej Grzegorczyk (\cite{Grzegorczyk:2012}).\par We present some {\em mathematical folklore} about representing formulas in ``Polish notation", that is, with operators of fixed arity prepended to their arguments. This notation, which was published by Jan {\L}ukasiewicz in~\cite{Lukasiewicz:1931}, eliminates the need for parentheses and is generally well suited for rigorous reasoning about syntactic properties of formulas.

- Taneli Huuskonen.
{G}rzegorczyk's Logics. {P}art {I},
Formalized Mathematics 23(3),
pages 177-187, 2015. MML Identifier:
GRZLOG_1

**Summary**: This article is the second in a series formalizing some results in my joint work with Prof.\ Joanna Goli\'nska-Pilarek (\cite{GPH:2012} and \cite{GPH:2015}) concerning a logic proposed by Prof. Andrzej Grzegorczyk (\cite{Grzegorczyk:2012}).\par This part presents the syntax and axioms of Grzegorczyk's {\em Logic of Descriptions} (LD) as originally proposed by him, as well as some theorems not depending on any semantic constructions. There are both some clear similarities and fundamental differences between LD and the non-Fregean logics introduced by Roman Suszko in~\cite{Suszko:1968}. In particular, we were inspired by Suszko's semantics for his non-Fregean logic~SCI, presented in~\cite{Suszko:1971}.

- Roland Coghetto.
{C}onvergent Filter Bases,
Formalized Mathematics 23(3),
pages 189-203, 2015. MML Identifier:
CARDFIL2

**Summary**: We are inspired by the work of Henri Cartan \cite{cartan1937a}, Bourbaki \cite{bourbaki2013general} (TG. I Filtres) and Claude Wagschal \cite{wagschal}. We define the base of filter, image filter, convergent filter bases, limit filter and the filter base of tails (fr: {\it filtre des sections}).

- Hiroyuki Okazaki and Yuichi Futa.
{P}olynomially Bounded Sequences and Polynomial Sequences,
Formalized Mathematics 23(3),
pages 205-213, 2015. MML Identifier:
ASYMPT_2

**Summary**: In this article, we formalize polynomially bounded sequences that plays an important role in computational complexity theory. Class P is a fundamental computational complexity class that contains all polynomial-time decision problems \cite{KleinbergTardos2005}, \cite{Knuth1997}. It takes polynomially bounded amount of computation time to solve polynomial-time decision problems by the deterministic Turing machine. Moreover we formalize polynomial sequences \cite{Barbeau2003}.

- Rafa{\l} Ziobro.
{F}ermat's {L}ittle {T}heorem via Divisibility of {N}ewton's Binomial,
Formalized Mathematics 23(3),
pages 215-229, 2015. MML Identifier:
NEWTON02

**Summary**: Solving equations in integers is an important part of the number theory \cite{sierpinski1956}. In many cases it can be conducted by the factorization of equation's elements, such as the Newton's binomial. The article introduces several simple formulas, which may facilitate this process. Some of them are taken from relevant books \cite{sierpinski1950}, \cite{gancarzewicz2000}. \par In the second section of the article, Fermat's Little Theorem is proved in a classical way, on the basis of divisibility of Newton's binomial. Although slightly redundant in its content (another proof of the theorem has earlier been included in \cite{EULER_2.ABS}), the article provides a good example, how the application of registrations could shorten the length of Mizar proofs \cite{caminati2013custom}, \cite{kornilowicz2013rewriting}.

- Keiko Narita, Noboru Endou and Yasunari Shidama.
{W}eak Convergence and Weak* Convergence,
Formalized Mathematics 23(3),
pages 231-241, 2015. MML Identifier:
DUALSP03

**Summary**: In this article, we deal with weak convergence on sequences in real normed spaces, and weak* convergence on sequences in dual spaces of real normed spaces. In the first section, we proved some topological properties of dual spaces of real normed spaces. We used these theorems for proofs of Section 3. In Section 2, we defined weak convergence and weak* convergence, and proved some properties. By \verb!RNS_Real! Mizar functor, real normed spaces as real number spaces already defined in the article \cite{DUALSP02.ABS}, we regarded sequences of real numbers as sequences of \verb!RNS_Real!. So we proved the last theorem in this section using the theorem (8) from \cite{LOPBAN_5.ABS}. In Section 3, we defined weak sequential compactness of real normed spaces. We showed some lemmas for the proof and proved the theorem of weak sequential compactness of reflexive real Banach spaces. We referred to \cite{yoshida:1980}, \cite{ReedSimon1972}, \cite{rudin1991functional} and \cite{Brezis2011} in the formalization.

- Keiko Narita, Noboru Endou and Yasunari Shidama.
{T}he Orthogonal Projection and the {R}iesz Representation Theorem,
Formalized Mathematics 23(3),
pages 243-252, 2015. MML Identifier:
DUALSP04

**Summary**: In this article, the orthogonal projection and the Riesz representation theorem are mainly formalized. In the first section, we defined the norm of elements on real Hilbert spaces, and defined Mizar functor \verb!RUSp2RNSp!, real normed spaces as real Hilbert spaces. By this definition, we regarded sequences of real Hilbert spaces as sequences of real normed spaces, and proved some properties of real Hilbert spaces. Furthermore, we defined the continuity and the Lipschitz the continuity of functionals on real Hilbert spaces. \par Referring to the article \cite{DUALSP01.ABS}, we also defined some definitions on real Hilbert spaces and proved some theorems for defining dual spaces of real Hilbert spaces. As to the properties of all definitions, we proved that they are equivalent properties of functionals on real normed spaces. In Sec. 2, by the definitions \cite{RUSUB_5.ABS}, we showed properties of the orthogonal complement. Then we proved theorems on the orthogonal decomposition of elements of real Hilbert spaces. They are the last two theorems of existence and uniqueness. In the third and final section, we defined the kernel of linear functionals on real Hilbert spaces. By the last three theorems, we showed the Riesz representation theorem, existence, uniqueness, and the property of the norm of bounded linear functionals on real Hilbert spaces. We referred to \cite{yoshida:1980}, \cite{Dax2002}, \cite{rudin1991functional} and \cite{Brezis2011} in the formalization.

- Noboru Endou.
{E}xtended Real-Valued Double Sequence and Its Convergence,
Formalized Mathematics 23(3),
pages 253-277, 2015. MML Identifier:
DBLSEQ_3

**Summary**: In this article we introduce the convergence of extended real-valued double sequences \cite{FOLLAND}, \cite{GARLING:1}. It is similar to our previous articles \cite{DBLSEQ_1.ABS}, \cite{DBLSEQ_2.ABS}. In addition, we also prove Fatou's lemma and the monotone convergence theorem for double sequences.

- Taneli Huuskonen.
{P}olish Notation,
Formalized Mathematics 23(3),
pages 161-176, 2015. MML Identifier:
POLNOT_1
- Number 4

- Roland Coghetto.
{S}ummable Family in a Commutative Group,
Formalized Mathematics 23(4),
pages 279-288, 2015. MML Identifier:
CARDFIL3

**Summary**: H{\"o}lzl et al. showed that it was possible to build ``a generic theory of limits based on filters" in Isabelle/HOL \cite{holzl2013type}, \cite{boldo2014formalization}. In this paper we present our formalization of this theory in Mizar \cite{Mizar-State-2015}. \par First, we compare the notions of the limit of a family indexed by a directed set, or a sequence, in a metric space \cite{FRECHET.ABS}, a real normed linear space \cite{NORMSP_1.ABS} and a linear topological space \cite{RLTOPSP1.ABS} with the concept of the limit of an image filter \cite{CARDFIL2.ABS}. \par Then, following Bourbaki \cite{bourbaki2007topologie}, \cite{bourbaki2013general} (TG.III, \S 5.1 {\it Familles sommables dans un groupe commutatif}), we conclude by defining the summable families in a commutative group (``additive notation" in \cite{GROUP_1A.ABS}), using the notion of filters.

- Roland Coghetto.
{T}opology from Neighbourhoods,
Formalized Mathematics 23(4),
pages 289-296, 2015. MML Identifier:
FINTOPO7

**Summary**: Using Mizar \cite{Mizar-State-2015}, and the formal topological space structure ({\tt FMT\_Space\_Str}) \cite{FINTOPO2.ABS}, we introduce the three {\tt U-FMT} conditions ({\tt U-FMT filter}, {\tt U-FMT with point} and {\tt U-FMT local}) similar to those $V_I$, $V_{II}$, $V_{III}$ and $V_{IV}$ of the proposition 2 in \cite{Bourbaki2013general}: \begin{quote} If to each element $x$ of a set $X$ there corresponds a set ${\cal{B}}(x)$ of subsets of $X$ such that the properties $V_I$, $V_{II}$, $V_{III}$ and $V_{IV}$ are satisfied, then there is a unique topological structure on $X$ such that, for each $x\in X$, ${\cal{B}}(x)$ is the set of neighborhoods of $x$ in this topology. \end{quote} We present a correspondence between a topological space and a space defined with the formal topological space structure with the three {\tt U-FMT} conditions called the topology from neighbourhoods. For the formalization, we were inspired by the works of Bourbaki \cite{bourbaki2007topologie} and Claude Wagschal \cite{wagschal}.

- Yuichi Futa, Hiroyuki Okazaki and Yasunari Shidama.
{T}orsion {P}art of {$\mathbb{Z}$}-module,
Formalized Mathematics 23(4),
pages 297-307, 2015. MML Identifier:
ZMODUL07

**Summary**: In this article, we formalize in Mizar \cite{Mizar-State-2015} the definition of ``torsion part" of $\mathbb Z$-module and its properties. We show $\mathbb Z$-module generated by the field of rational numbers as an example of torsion-free non free $\mathbb Z$-modules. We also formalize the rank-nullity theorem over finite-rank free $\mathbb Z$-modules (previously formalized in \cite{RANKNULL.ABS}). $\mathbb Z$-module is necessary for lattice problems, LLL (Lenstra, Lenstra and Lov\'asz) base reduction algorithm \cite{LLL} and cryptographic systems with lattices \cite{LATTICE2002}.

- Noboru Endou.
{C}onstruction of Measure from Semialgebra of Sets,
Formalized Mathematics 23(4),
pages 309-323, 2015. MML Identifier:
MEASURE9

**Summary**: In our previous article \cite{MEASURE8.ABS}, we showed complete additivity as a condition for extension of a measure. However, this condition premised the existence of a $\sigma$-field and the measure on it. In general, the existence of the measure on $\sigma$-field is not obvious. On the other hand, the proof of existence of a measure on a semialgebra is easier than in the case of a $\sigma$-field. Therefore, in this article we define a measure ({\tt pre-measure}) on a semialgebra and extend it to a measure on a $\sigma$-field. Furthermore, we give a $\sigma$-measure as an extension of the measure on a $\sigma$-field. We follow \cite{Halmos74}, \cite{Bogachev2006}, and \cite{Rao2004}.

- Ievgen Ivanov, Mykola Nikitchenko and Uri Abraham.
{E}vent-Based Proof of the Mutual Exclusion Property of {P}eterson's Algorithm,
Formalized Mathematics 23(4),
pages 325-331, 2015. MML Identifier:
PETERSON

**Summary**: Proving properties of distributed algorithms is still a highly challenging problem and various approaches that have been proposed to tackle it \cite{Abraham1999} can be roughly divided into state-based and event-based proofs. Informally speaking, state-based approaches define the behavior of a distributed algorithm as a set of sequences of memory states during its executions, while event-based approaches treat the behaviors by means of events which are produced by the executions of an algorithm. Of course, combined approaches are also possible. \par Analysis of the literature \cite{Abraham1999}, \cite{chandy1988}, \cite{Pratt1986}, \cite{lamport1986}, \cite{raynal991}, \cite{Ridge2006}, \cite{Ridge2007} shows that state-based approaches are more widely used than event-based approaches for proving properties of algorithms, and the difficulties in the event-based approach are often emphasized. We believe, however, that there is a certain naturalness and intuitive content in event-based proofs of correctness of distributed algorithms that makes this approach worthwhile. Besides, state-based proofs of correctness of distributed algorithms are usually applicable only to discrete-time models of distributed systems and cannot be easily adapted to the continuous time case which is important in the domain of cyber-physical systems. On the other hand, event-based proofs can be readily applied to continuous-time / hybrid models of distributed systems. \par In the paper \cite{AbrIvNikitch2011} we presented a compositional approach to reasoning about behavior of distributed systems in terms of events. Compositionality here means (informally) that semantics and properties of a program is determined by semantics of processes and process communication mechanisms. We demonstrated the proposed approach on a proof of the mutual exclusion property of the Peterson's algorithm \cite{Peterson1981}. We have also demonstrated an application of this approach for proving the mutual exclusion property in the setting of continuous-time models of cyber-physical systems in \cite{IvanovNikitchAbr2014}. \par Using Mizar \cite{Mizar-State-2015}, in this paper we give a formal proof of the mutual exclusion property of the Peterson's algorithm in Mizar on the basis of the event-based approach proposed in \cite{AbrIvNikitch2011}. Firstly, we define an event-based model of a shared-memory distributed system as a multi-sorted algebraic structure in which sorts are events, processes, locations (i.e. addresses in the shared memory), traces (of the system). The operations of this structure include a binary precedence relation $\leq$ on the set of events which turns it into a linear preorder (events are considered simultaneous, if ${e_1}\leq{e_2}$ and ${e_2}\leq{e_1}$), special predicates which check if an event occurs in a given process or trace, predicates which check if an event causes the system to read from or write to a given memory location, and a special partial function ``{\tt val of}" on events which gives the value associated with a memory read or write event (i.e. a value which is written or is read in this event) \cite{AbrIvNikitch2011}. Then we define several natural consistency requirements (axioms) for this structure which must hold in every distributed system, e.g. each event occurs in some process, etc. (details are given in \cite{AbrIvNikitch2011}). \par After this we formulate and prove the main theorem about the mutual exclusion property of the Peterson's algorithm in an arbitrary consistent algebraic structure of events. Informally, the main theorem states that if a system consists of two processes, and in some trace there occur two events $e_1$ and $e_2$ in different processes and each of these events is preceded by a series of three special events (in the same process) guaranteed by execution of the Peterson's algorithm (setting the flag of the current process, writing the identifier of the opposite process to the ``turn" shared variable, and reading zero from the flag of the opposite process or reading the identifier of the current process from the ``turn" variable), and moreover, if neither process writes to the flag of the opposite process or writes its own identifier to the ``turn" variable, then either the events $e_1$ and $e_2$ coincide, or they are not simultaneous (mutual exclusion property).

- Christoph Schwarzweller and Artur Korni{\l}owicz.
{C}haracteristic of Rings. {P}rime Fields,
Formalized Mathematics 23(4),
pages 333-349, 2015. MML Identifier:
RING_3

**Summary**: The notion of the characteristic of rings and its basic properties are formalized \cite{Jacobson2009}, \cite{Waerden2003}, \cite{Luneburg1999}. Classification of prime fields in terms of isomorphisms with appropriate fields ($\mathbb{Q}$ or $\mathbb{Z}/p$) are presented. To facilitate reasonings within the field of rational numbers, values of numerators and denominators of basic operations over rationals are computed.

- Marco Riccardi.
{E}xponential Objects,
Formalized Mathematics 23(4),
pages 351-369, 2015. MML Identifier:
CAT_8

**Summary**: In the first part of this article we formalize the concepts of terminal and initial object, categorical product \cite{Borceaux} and natural transformation within a free-object category \cite{Adamek:2009}. In particular, we show that this definition of natural transformation is equivalent to the standard definition \cite{MacLane:1}. Then we introduce the exponential object using its universal property and we show the isomorphism between the exponential object of categories and the functor category \cite{Lawvere:1963}.

- Hiroyuki Okazaki.
{A}lgebra of Polynomially Bounded Sequences and Negligible Functions,
Formalized Mathematics 23(4),
pages 371-378, 2015. MML Identifier:
ASYMPT_3

**Summary**: In this article we formalize negligible functions that play an essential role in cryptology \cite{GOLDREICH}, \cite{BELLARE}. Generally, a cryptosystem is secure if the probability of succeeding any attacks against the cryptosystem is negligible. First, we formalize the algebra of polynomially bounded sequences \cite{ASYMPT_2.ABS}. Next, we formalize negligible functions and prove the set of negligible functions is a subset of the algebra of polynomially bounded sequences. Moreover, we then introduce equivalence relation between polynomially bounded sequences, using negligible functions.

- Mariusz Giero.
{P}ropositional Linear Temporal Logic with~Initial Validity Semantics,
Formalized Mathematics 23(4),
pages 379-386, 2015. MML Identifier:
LTLAXIO5

**Summary**: In the article \cite{LTLAXIO1.ABS} a formal system for Propositional Linear Temporal Logic (in short LTLB) with normal semantics is introduced. The language of this logic consists of ``until" operator in a very strict version. The very strict ``until" operator enables to express all other temporal operators.\par In this article we construct a formal system for LTLB with the initial semantics \cite{KroMer}. Initial semantics means that we define the validity of the formula in a model as satisfaction in the initial state of model while normal semantics means that we define the validity as satisfaction in all states of model. We prove the Deduction Theorem, and the soundness and completeness of the introduced formal system. We also prove some theorems to compare both formal systems, i.e., the one introduced in the article \cite{LTLAXIO1.ABS} and the one introduced in this article. \par Formal systems for temporal logics are applied in the verification of computer programs. In order to carry out the verification one has to derive an appropriate formula within a selected formal system. The formal systems introduced in \cite{LTLAXIO1.ABS} and in this article can be used to carry out such verifications in Mizar \cite{Mizar-State-2015}.

- Adam Grabowski.
{S}tone Lattices,
Formalized Mathematics 23(4),
pages 387-396, 2015. MML Identifier:
LATSTONE

**Summary**: The article continues the formalization of the lattice theory (as structures with two binary operations, not in terms of ordering relations). In the paper, the notion of a pseudocomplement in a lattice is formally introduced in Mizar, and based on this we define the notion of the skeleton and the set of dense elements in a pseudocomplemented lattice, giving the meet-decomposition of arbitrary element of a lattice as the infimum of two elements: one belonging to the skeleton, and the other which is dense. \par The core of the paper is of course the idea of Stone identity $$a^{\star} \sqcup a^{\star\star} = \top,$$ which is fundamental for us: Stone lattices are those lattices $L$, which are distributive, bounded, and satisfy Stone identity for all elements $a\in L.$ Stone algebras were introduced by Gr\"atzer and Schmidt in \cite{Gratzer:1957}. Of course, the pseudocomplement is unique (if exists), so in a pseudcomplemented lattice we defined $a^{\star}$ as the Mizar functor (unary operation mapping every element to its pseudocomplement). In Section 2 we prove formally a collection of ordinary properties of pseudocomplemented lattices. \par All Boolean lattices are Stone, and a natural example of the lattice which is Stone, but not Boolean, is the lattice of all natural divisors of $p^2$ for arbitrary prime number $p$ (Section 6). At the end we formalize the notion of the Stone lattice $B^{[2]}$ (of pairs of elements $a,b$ of $B$ such that $a\leq b$) constructed as a sublattice of $B^2$, where $B$ is arbitrary Boolean algebra (and we describe skeleton and the set of dense elements in such lattices). In a natural way, we deal with Cartesian product of pseudocomplemented lattices. \par Our formalization was inspired by \cite{Gratzer2011}, and is an important step in formalizing Jouni J{\"a}rvinen \emph{Lattice theory for rough sets} \cite{Jarvinen:2007}, so it follows rather the latter paper. We deal essentially with Section 4.3, pages 423--426. The description of handling complemented structures in Mizar \cite{Mizar-State-2015} can be found in \cite{GrabowskiJAR40}. The current article together with \cite{NELSON_1.ABS} establishes the formal background for algebraic structures which are important for \cite{GrabowskiAssisted:2005}, \cite{GrabowskiPerspective:2007} by means of mechanisms of merging theories as described in \cite{GrabowskiFI:2014}.

- Roland Coghetto.
{S}ummable Family in a Commutative Group,
Formalized Mathematics 23(4),
pages 279-288, 2015. MML Identifier:
CARDFIL3
**Volume 24, 2016**- Number 1

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