Volume 9, 1997

University of Bialystok

Copyright (c) 1997 Association of Mizar Users

**Andrzej Trybulec**- University of Bialystok
**Yatsuka Nakamura**- Shinshu University, Nagano

- 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.

- General preliminaries
- Preliminaries (general topology)
- Preliminaries (the topology of the plane)
- Special points of a compact non empty subset of the plane
- Subsets of the plane that are neither vertical nor horizontal
- A special sequence related to a compact non empty subset of the plane
- Rectangular finite suequences of the points of the plane
- Jordan property

- [1]
Grzegorz Bancerek.
Cardinal numbers.
*Journal of Formalized Mathematics*, 1, 1989. - [2]
Grzegorz Bancerek.
The ordinal numbers.
*Journal of Formalized Mathematics*, 1, 1989. - [3]
Grzegorz Bancerek and Krzysztof Hryniewiecki.
Segments of natural numbers and finite sequences.
*Journal of Formalized Mathematics*, 1, 1989. - [4]
Jozef Bialas.
Group and field definitions.
*Journal of Formalized Mathematics*, 1, 1989. - [5]
Czeslaw Bylinski.
Functions and their basic properties.
*Journal of Formalized Mathematics*, 1, 1989. - [6]
Czeslaw Bylinski.
Functions from a set to a set.
*Journal of Formalized Mathematics*, 1, 1989. - [7]
Czeslaw Bylinski.
Some basic properties of sets.
*Journal of Formalized Mathematics*, 1, 1989. - [8]
Czeslaw Bylinski and Yatsuka Nakamura.
Special polygons.
*Journal of Formalized Mathematics*, 7, 1995. - [9]
Czeslaw Bylinski and Piotr Rudnicki.
Bounding boxes for compact sets in $\calE^2$.
*Journal of Formalized Mathematics*, 9, 1997. - [10]
Agata Darmochwal.
Compact spaces.
*Journal of Formalized Mathematics*, 1, 1989. - [11]
Agata Darmochwal.
The Euclidean space.
*Journal of Formalized Mathematics*, 3, 1991. - [12]
Agata Darmochwal and Yatsuka Nakamura.
The topological space $\calE^2_\rmT$. Arcs, line segments and special polygonal arcs.
*Journal of Formalized Mathematics*, 3, 1991. - [13]
Krzysztof Hryniewiecki.
Basic properties of real numbers.
*Journal of Formalized Mathematics*, 1, 1989. - [14]
Katarzyna Jankowska.
Matrices. Abelian group of matrices.
*Journal of Formalized Mathematics*, 3, 1991. - [15]
Jaroslaw Kotowicz.
Convergent real sequences. Upper and lower bound of sets of real numbers.
*Journal of Formalized Mathematics*, 1, 1989. - [16]
Jaroslaw Kotowicz.
Monotone real sequences. Subsequences.
*Journal of Formalized Mathematics*, 1, 1989. - [17]
Yatsuka Nakamura and Czeslaw Bylinski.
Extremal properties of vertices on special polygons, part I.
*Journal of Formalized Mathematics*, 6, 1994. - [18]
Yatsuka Nakamura and Jaroslaw Kotowicz.
The Jordan's property for certain subsets of the plane.
*Journal of Formalized Mathematics*, 4, 1992. - [19]
Yatsuka Nakamura and Andrzej Trybulec.
Decomposing a Go-Board into cells.
*Journal of Formalized Mathematics*, 7, 1995. - [20]
Beata Padlewska.
Connected spaces.
*Journal of Formalized Mathematics*, 1, 1989. - [21]
Beata Padlewska and Agata Darmochwal.
Topological spaces and continuous functions.
*Journal of Formalized Mathematics*, 1, 1989. - [22]
Konrad Raczkowski and Pawel Sadowski.
Topological properties of subsets in real numbers.
*Journal of Formalized Mathematics*, 2, 1990. - [23]
Andrzej Trybulec.
Tarski Grothendieck set theory.
*Journal of Formalized Mathematics*, Axiomatics, 1989. - [24]
Andrzej Trybulec.
Left and right component of the complement of a special closed curve.
*Journal of Formalized Mathematics*, 7, 1995. - [25]
Andrzej Trybulec.
On the decomposition of finite sequences.
*Journal of Formalized Mathematics*, 7, 1995. - [26]
Andrzej Trybulec.
Subsets of real numbers.
*Journal of Formalized Mathematics*, Addenda, 2003. - [27]
Andrzej Trybulec and Czeslaw Bylinski.
Some properties of real numbers operations: min, max, square, and square root.
*Journal of Formalized Mathematics*, 1, 1989. - [28]
Wojciech A. Trybulec.
Pigeon hole principle.
*Journal of Formalized Mathematics*, 2, 1990. - [29]
Zinaida Trybulec.
Properties of subsets.
*Journal of Formalized Mathematics*, 1, 1989. - [30]
Edmund Woronowicz.
Relations and their basic properties.
*Journal of Formalized Mathematics*, 1, 1989.

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