FORMALIZED MATHEMATICS
Vol.2, No. 4, September–October 1991
Universit´e Catholique de Louvain
The Euclidean Space
Agata Darmochwa l
Warsaw University
Bia l ystok
Summary. The general definition of Euclidean Space.
MML Identifier:EUCLID .
The papers [14],[6],[9],[8],[12],[1],[5],[10],[3],[13],[4],[15],[16],[7],[11], and [2]provide the notation and terminology for this paper. In the sequel k , n denote natural numbers and r denotes a real number. Let us consider n . The functor R n yields a non-empty set of finite sequences of and is defined as follows:
(Def.1)R n =n .
In the sequel x will denote a finite sequence of elements of . The function ||from into is defined as follows:
(Def.2)for every r holds ||(r ) =|r |.
Let us consider x . The functor |x |yields a finite sequence of elements of and is defined as follows:
(Def.3)|x |=||·x .
Let us consider n . The functor 0, ... , 0Let us consider n . The functor 0, ... , 0
n yields a finite sequence of elements of and is defined by:
(Def.4) 0, ... , 0
n =n − → 0qua a real number .
Let us consider n . Then 0, ... , 0
n is an element of R n .
In the sequel x , x 1, x 2, y denote elements of R n . One can prove the following proposition
(1)x is an element of n .
601 c 1991Fondation Philippe le Hodey ISSN 0777–4028
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