@MISC{Darmochwał_summary.the, author = {Agata Darmochwał}, title = {Summary. The general definition of Euclidean Space.}, year = {} }

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Abstract

the notation and terminology for this paper. In this paper k, n denote natural numbers and r denotes a real number. Let us consider n. The functorR n yields a non empty set of finite sequences of R and is defined as follows: (Def. 1) R n = R n. In the sequel x denotes a finite sequence of elements of R. The function |�|R from R into R is defined by: (Def. 2) For every r holds |�|R(r) = |r|. Let us consider x. The functor |x | yielding a finite sequence of elements of R is defined as follows: (Def. 3) |x | = |�|R · x. Let us consider n. The functor 〈0,...,0 〉 yields a finite sequence of elements of R and is defined n as follows: (Def. 4) 〈0,...,0 〉 = n ↦ → (0 qua real number). n Let us consider n. Then 〈0,...,0 〉 is an element ofR n. n In the sequel x, x1, x2, y denote elements ofR n. Let us consider n, x. Then −x is an element ofR n. Let us consider n, x, y. Then x+y is an element ofR n. Then x − y is an element ofR n. Let us consider n, let r be a real number, and let us consider x. Then r · x is an element ofR n. Let us consider n, x. Then |x | is an element of R n. Let us consider n, x. Then 2 x is an element of R n. Let x be a finite sequence of elements of R. The functor |x | yields a real number and is defined as follows: (Def. 5) |x | = � ∑ 2 x. 1 c ○ Association of Mizar Users One can prove the following propositions: (2) 1 lenx = n. (3) domx = Segn. (4) x(k) ∈ R. THE EUCLIDEAN SPACE 2 (5) If for every k such that k ∈ Segn holds x1(k) = x2(k), then x1 = x2. (6) If r = x(k), then |x|(k) = |r|. (7) |〈0,...,0 〉 | = n ↦ → (0 qua real number).