Journal of Formalized Mathematics
Volume 11, 1999
University of Bialystok
Copyright (c) 1999 Association of Mizar Users

## The Definition of the Riemann Definite Integral and some Related Lemmas

Noboru Endou
Shinshu University, Nagano
Artur Kornilowicz
University of Bialystok
This paper was written while the second author visited Shinshu University, winter 1999.

### 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.

#### MML Identifier: INTEGRA1

The terminology and notation used in this paper have been introduced in the following articles                              

#### Contents (PDF format)

1. Definition of Closed Interval and its Properties
2. Definition of Division of Closed Interval and its Properties
3. Definitions of Integrability and Related Topics
4. Real Function's Properties
5. Characteristic Function's Properties
6. Some Properties of Darboux Sum

#### Bibliography

 Grzegorz Bancerek. Cardinal numbers. Journal of Formalized Mathematics, 1, 1989.
 Grzegorz Bancerek. The ordinal numbers. Journal of Formalized Mathematics, 1, 1989.
 Grzegorz Bancerek and Krzysztof Hryniewiecki. Segments of natural numbers and finite sequences. Journal of Formalized Mathematics, 1, 1989.
 Czeslaw Bylinski. Functions and their basic properties. Journal of Formalized Mathematics, 1, 1989.
 Czeslaw Bylinski. Functions from a set to a set. Journal of Formalized Mathematics, 1, 1989.
 Czeslaw Bylinski. Partial functions. Journal of Formalized Mathematics, 1, 1989.
 Czeslaw Bylinski. Some basic properties of sets. Journal of Formalized Mathematics, 1, 1989.
 Czeslaw Bylinski. Finite sequences and tuples of elements of a non-empty sets. Journal of Formalized Mathematics, 2, 1990.
 Czeslaw Bylinski. The sum and product of finite sequences of real numbers. Journal of Formalized Mathematics, 2, 1990.
 Czeslaw Bylinski and Piotr Rudnicki. Bounding boxes for compact sets in \$\calE^2\$. Journal of Formalized Mathematics, 9, 1997.
 Czeslaw Bylinski and Andrzej Trybulec. Complex spaces. Journal of Formalized Mathematics, 2, 1990.
 Agata Darmochwal. Finite sets. Journal of Formalized Mathematics, 1, 1989.
 Agata Darmochwal and Yatsuka Nakamura. The topological space \$\calE^2_\rmT\$. Arcs, line segments and special polygonal arcs. Journal of Formalized Mathematics, 3, 1991.
 Krzysztof Hryniewiecki. Basic properties of real numbers. Journal of Formalized Mathematics, 1, 1989.
 Jaroslaw Kotowicz. Convergent real sequences. Upper and lower bound of sets of real numbers. Journal of Formalized Mathematics, 1, 1989.
 Jaroslaw Kotowicz. Real sequences and basic operations on them. Journal of Formalized Mathematics, 1, 1989.
 Jaroslaw Kotowicz. Partial functions from a domain to a domain. Journal of Formalized Mathematics, 2, 1990.
 Jaroslaw Kotowicz. Partial functions from a domain to the set of real numbers. Journal of Formalized Mathematics, 2, 1990.
 Jaroslaw Kotowicz. Functions and finite sequences of real numbers. Journal of Formalized Mathematics, 5, 1993.
 Jaroslaw Kotowicz and Yatsuka Nakamura. Introduction to Go-Board --- part I. Journal of Formalized Mathematics, 4, 1992.
 Eugeniusz Kusak, Wojciech Leonczuk, and Michal Muzalewski. Abelian groups, fields and vector spaces. Journal of Formalized Mathematics, 1, 1989.
 Yatsuka Nakamura and Roman Matuszewski. Reconstructions of special sequences. Journal of Formalized Mathematics, 8, 1996.
 Yatsuka Nakamura, Piotr Rudnicki, Andrzej Trybulec, and Pauline N. Kawamoto. Preliminaries to circuits, I. Journal of Formalized Mathematics, 6, 1994.
 Konrad Raczkowski and Pawel Sadowski. Topological properties of subsets in real numbers. Journal of Formalized Mathematics, 2, 1990.
 Andrzej Trybulec. Binary operations applied to functions. Journal of Formalized Mathematics, 1, 1989.
 Andrzej Trybulec. Tarski Grothendieck set theory. Journal of Formalized Mathematics, Axiomatics, 1989.
 Andrzej Trybulec. Subsets of real numbers. Journal of Formalized Mathematics, Addenda, 2003.
 Wojciech A. Trybulec. Pigeon hole principle. Journal of Formalized Mathematics, 2, 1990.
 Zinaida Trybulec. Properties of subsets. Journal of Formalized Mathematics, 1, 1989.
 Edmund Woronowicz. Relations defined on sets. Journal of Formalized Mathematics, 1, 1989.