Journal of Formalized Mathematics
Volume 2, 1990
University of Bialystok
Copyright (c) 1990 Association of Mizar Users

## The Sum and Product of Finite Sequences of Real Numbers

Czeslaw Bylinski
Warsaw University, Bialystok
Supported by RPBP.III-24.C1.

### 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  are used. Then the fact that certain properties are satisfied by those operations is demonstrated directly from . 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 . 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.

#### MML Identifier: RVSUM_1

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

Contents (PDF format)

#### Bibliography

 Grzegorz Bancerek. The fundamental properties of natural 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. Binary operations. 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. Binary operations applied to finite sequences. Journal of Formalized Mathematics, 2, 1990.
 Czeslaw Bylinski. Finite sequences and tuples of elements of a non-empty sets. Journal of Formalized Mathematics, 2, 1990.
 Czeslaw Bylinski. Semigroup operations on finite subsets. Journal of Formalized Mathematics, 2, 1990.
 Krzysztof Hryniewiecki. Basic properties of real numbers. Journal of Formalized Mathematics, 1, 1989.
 Jaroslaw Kotowicz. Real sequences and basic operations on them. Journal of Formalized Mathematics, 1, 1989.
 Eugeniusz Kusak, Wojciech Leonczuk, and Michal Muzalewski. Abelian groups, fields and vector spaces. Journal of Formalized Mathematics, 1, 1989.
 Andrzej Trybulec. Binary operations applied to functions. Journal of Formalized Mathematics, 1, 1989.
 Andrzej Trybulec. Semilattice operations on finite subsets. 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.
 Andrzej Trybulec and Czeslaw Bylinski. Some properties of real numbers operations: min, max, square, and square root. Journal of Formalized Mathematics, 1, 1989.
 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 and their basic properties. Journal of Formalized Mathematics, 1, 1989.

Received May 11, 1990

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