Volume 4, 1992

University of Bialystok

Copyright (c) 1992 Association of Mizar Users

**Michal Muzalewski**- Warsaw University, Bialystok

- We shall describe $n$-dimensional spaces with the reper operation [7, pages 72-79]. An inspiration to such approach comes from the monograph [9] and so-called Leibniz program. Let us recall that the Leibniz program is a program of algebraization of geometry using purely geometric notions. Leibniz formulated his program in opposition to algebraization method developed by Descartes. The Euclidean geometry in Szmielew's approach [0] {SZMIELEW:1} is a theory of structures $\langle S$; $\parallel, \oplus, O \rangle$, where $\langle S$; $\parallel, \oplus, O \rangle$ is Desarguesian midpoint plane and $O \subseteq S\times S\times S$ is the relation of equi-orthogonal basis. Points $o, p, q$ are in relation $O$ if they form an isosceles triangle with the right angle in vertex $a$. If we fix vertices $a, p$, then there exist exactly two points $q, q'$ such that $O(apq)$, $O(apq')$. Moreover $q \oplus q' = a$. In accordance with the Leibniz program we replace the relation of equi-orthogonal basis by a binary operation $\ast : S\times S \rightarrow S$, called the reper operation. A standard model for the Euclidean geometry in the above sense is the oriented plane over the field of real numbers with the reper operations $\ast$ defined by the condition: $a \ast b = q$ iff the point $q$ is the result of rotating of $p$ about right angle around the center $a$.

- Substitutions in tuples
- Reper Algebra Structure and its Properties
- Reper Algebra and its Atlas

- [1]
Grzegorz Bancerek.
The fundamental properties of natural 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 and their basic properties.
*Journal of Formalized Mathematics*, 1, 1989. - [4]
Czeslaw Bylinski.
Functions from a set to a set.
*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]
Michal Muzalewski.
Midpoint algebras.
*Journal of Formalized Mathematics*, 1, 1989. - [7] Michal Muzalewski. \em Foundations of Metric-Affine Geometry. Dzial Wydawnictw Filii UW w Bialymstoku, Filia UW w Bialymstoku, 1990.
- [8]
Michal Muzalewski.
Atlas of midpoint algebra.
*Journal of Formalized Mathematics*, 3, 1991. - [9] Wanda Szmielew. \em From Affine to Euclidean Geometry, volume 27. PWN -- D.Reidel Publ. Co., Warszawa -- Dordrecht, 1983.
- [10]
Andrzej Trybulec.
Tarski Grothendieck set theory.
*Journal of Formalized Mathematics*, Axiomatics, 1989. - [11]
Wojciech A. Trybulec.
Vectors in real linear space.
*Journal of Formalized Mathematics*, 1, 1989. - [12]
Zinaida Trybulec.
Properties of subsets.
*Journal of Formalized Mathematics*, 1, 1989.

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