Volume 12, 2000

University of Bialystok

Copyright (c) 2000 Association of Mizar Users

**Noboru Endou**- Shinshu University, Nagano
**Katsumi Wasaki**- Shinshu University, Nagano
**Yasunari Shidama**- Shinshu University, Nagano

- 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.

- Some Useful Properties of Finite Sequence
- Integrability for Partial Function of ${\Bbb R}$, ${\Bbb R}$
- Integrability for Continuous Function

- [1]
Grzegorz Bancerek.
The ordinal numbers.
*Journal of Formalized Mathematics*, 1, 1989. - [2]
Grzegorz Bancerek and Krzysztof Hryniewiecki.
Segments of natural numbers and finite sequences.
*Journal of Formalized Mathematics*, 1, 1989. - [3]
Czeslaw Bylinski.
Functions from a set to a set.
*Journal of Formalized Mathematics*, 1, 1989. - [4]
Czeslaw Bylinski.
Partial functions.
*Journal of Formalized Mathematics*, 1, 1989. - [5]
Czeslaw Bylinski.
Finite sequences and tuples of elements of a non-empty sets.
*Journal of Formalized Mathematics*, 2, 1990. - [6]
Czeslaw Bylinski.
The sum and product of finite sequences of real numbers.
*Journal of Formalized Mathematics*, 2, 1990. - [7]
Czeslaw Bylinski and Piotr Rudnicki.
Bounding boxes for compact sets in $\calE^2$.
*Journal of Formalized Mathematics*, 9, 1997. - [8]
Noboru Endou and Artur Kornilowicz.
The definition of the Riemann definite integral and some related lemmas.
*Journal of Formalized Mathematics*, 11, 1999. - [9]
Krzysztof Hryniewiecki.
Basic properties of real numbers.
*Journal of Formalized Mathematics*, 1, 1989. - [10]
Jaroslaw Kotowicz.
Convergent real sequences. Upper and lower bound of sets of real numbers.
*Journal of Formalized Mathematics*, 1, 1989. - [11]
Jaroslaw Kotowicz.
Convergent sequences and the limit of sequences.
*Journal of Formalized Mathematics*, 1, 1989. - [12]
Jaroslaw Kotowicz.
Real sequences and basic operations on them.
*Journal of Formalized Mathematics*, 1, 1989. - [13]
Jaroslaw Kotowicz.
Partial functions from a domain to the set of real numbers.
*Journal of Formalized Mathematics*, 2, 1990. - [14]
Jaroslaw Kotowicz.
Properties of real functions.
*Journal of Formalized Mathematics*, 2, 1990. - [15]
Eugeniusz Kusak, Wojciech Leonczuk, and Michal Muzalewski.
Abelian groups, fields and vector spaces.
*Journal of Formalized Mathematics*, 1, 1989. - [16]
Konrad Raczkowski and Pawel Sadowski.
Real function continuity.
*Journal of Formalized Mathematics*, 2, 1990. - [17]
Konrad Raczkowski and Pawel Sadowski.
Real function differentiability.
*Journal of Formalized Mathematics*, 2, 1990. - [18]
Konrad Raczkowski and Pawel Sadowski.
Topological properties of subsets in real numbers.
*Journal of Formalized Mathematics*, 2, 1990. - [19]
Andrzej Trybulec.
Tarski Grothendieck set theory.
*Journal of Formalized Mathematics*, Axiomatics, 1989. - [20]
Andrzej Trybulec.
Subsets of real numbers.
*Journal of Formalized Mathematics*, Addenda, 2003. - [21]
Zinaida Trybulec.
Properties of subsets.
*Journal of Formalized Mathematics*, 1, 1989. - [22]
Edmund Woronowicz.
Relations defined on sets.
*Journal of Formalized Mathematics*, 1, 1989.

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