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

Scalar Multiple of Riemann Definite Integral


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

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 [11].

MML Identifier: INTEGRA2

The terminology and notation used in this paper have been introduced in the following articles [20] [22] [3] [21] [12] [2] [5] [23] [13] [7] [6] [9] [4] [15] [8] [11] [14] [17] [18] [19] [10] [1] [16]

Contents (PDF format)

  1. Lemmas of Finite Sequence
  2. Scalar Product of Real Subset
  3. Scalar Multiple of Integral
  4. Monotoneity of Integral
  5. Definition of Division Sequence

Bibliography

[1] Grzegorz Bancerek. Cardinal numbers. Journal of Formalized Mathematics, 1, 1989.
[2] Grzegorz Bancerek. The fundamental properties of natural numbers. Journal of Formalized Mathematics, 1, 1989.
[3] Grzegorz Bancerek. The ordinal numbers. Journal of Formalized Mathematics, 1, 1989.
[4] Grzegorz Bancerek and Krzysztof Hryniewiecki. Segments of natural numbers and finite sequences. 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. Partial functions. Journal of Formalized Mathematics, 1, 1989.
[8] Czeslaw Bylinski. The sum and product of finite sequences of real numbers. Journal of Formalized Mathematics, 2, 1990.
[9] Czeslaw Bylinski and Piotr Rudnicki. Bounding boxes for compact sets in $\calE^2$. Journal of Formalized Mathematics, 9, 1997.
[10] Agata Darmochwal. Finite sets. Journal of Formalized Mathematics, 1, 1989.
[11] Noboru Endou and Artur Kornilowicz. The definition of the Riemann definite integral and some related lemmas. Journal of Formalized Mathematics, 11, 1999.
[12] Krzysztof Hryniewiecki. Basic properties of real numbers. Journal of Formalized Mathematics, 1, 1989.
[13] Jaroslaw Kotowicz. Convergent real sequences. Upper and lower bound of sets of real numbers. Journal of Formalized Mathematics, 1, 1989.
[14] Jaroslaw Kotowicz. Real sequences and basic operations on them. Journal of Formalized Mathematics, 1, 1989.
[15] Jaroslaw Kotowicz. Partial functions from a domain to the set of real numbers. Journal of Formalized Mathematics, 2, 1990.
[16] Jaroslaw Kotowicz. Functions and finite sequences of real numbers. Journal of Formalized Mathematics, 5, 1993.
[17] Yatsuka Nakamura and Roman Matuszewski. Reconstructions of special sequences. Journal of Formalized Mathematics, 8, 1996.
[18] Yatsuka Nakamura, Piotr Rudnicki, Andrzej Trybulec, and Pauline N. Kawamoto. Preliminaries to circuits, I. Journal of Formalized Mathematics, 6, 1994.
[19] Konrad Raczkowski and Pawel Sadowski. Topological properties of subsets in real numbers. Journal of Formalized Mathematics, 2, 1990.
[20] Andrzej Trybulec. Tarski Grothendieck set theory. Journal of Formalized Mathematics, Axiomatics, 1989.
[21] Andrzej Trybulec. Subsets of real numbers. Journal of Formalized Mathematics, Addenda, 2003.
[22] Zinaida Trybulec. Properties of subsets. Journal of Formalized Mathematics, 1, 1989.
[23] Edmund Woronowicz. Relations defined on sets. Journal of Formalized Mathematics, 1, 1989.

Received December 7, 1999

[ Download a postscript version, MML identifier index, Mizar home page]