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Subsequences of Almost, Weakly and Poorly Onetoone Finite Sequences 1
"... notation for this paper. In this paper n is a natural number. The following three propositions are true: if: (1) For every finite sequence f of elements of E 2 T and for every point p of E2 T such that p ∈ � L(f) holds len⇃p, f ≥ 1. (2) For every non empty finite sequence f of elements of E2 T and ..."
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notation for this paper. In this paper n is a natural number. The following three propositions are true: if: (1) For every finite sequence f of elements of E 2 T and for every point p of E2 T such that p ∈ � L(f) holds len⇃p, f ≥ 1. (2) For every non empty finite sequence f of elements of E2 T and for every point p of E2 T holds len⇂f,p ≥ 1. (3) For every finite sequence f of elements of E2 T and for all points p, q of holds ⇃⇂p, f, q � = ∅. E 2 T Let x be a set. One can check that 〈x 〉 is onetoone. Let f be a finite sequence. We say that f is almost onetoone if and only (Def. 1) For all natural numbers i, j such that i ∈ domf and j ∈ domf and i � = 1 or j � = lenf and i � = lenf or j � = 1 and f(i) = f(j) holds i = j. if: Let f be a finite sequence. We say that f is weakly onetoone if and only (Def. 2) For every natural number i such that 1 ≤ i and i < len f holds f(i) �= f(i + 1).
Some Properties of Isomorphism between Relational Structures. On the Product of Topological Spaces
, 2002
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Partial differentiation on normed linear spaces Rn.
 Formalized Mathematics,
, 2007
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Niemytzki plane  an example of Tychonoff . . .
, 2005
"... We continue Mizar formalization of General Topology according to the book [20] by Engelking. Niemytzki plane is defined as halfplane y ≥ 0 with topology introduced by a neighborhood system. Niemytzki plane is not T4. Next, the definition of Tychonoff space is given. The characterization of Tychonoff ..."
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We continue Mizar formalization of General Topology according to the book [20] by Engelking. Niemytzki plane is defined as halfplane y ≥ 0 with topology introduced by a neighborhood system. Niemytzki plane is not T4. Next, the definition of Tychonoff space is given. The characterization of Tychonoff space by prebasis and the fact that Tychonoff spaces are between T3 and T4 is proved. The final result is that Niemytzki plane is also a Tychonoff space.
The definition of finite sequences and matrices of . . .
, 2006
"... In this article, we first define finite sequences of probability distribution and matrices of joint probability and conditional probability. We discuss also the concept of marginal probability. Further, we describe some theorems of matrices of real elements including quadratic form. ..."
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In this article, we first define finite sequences of probability distribution and matrices of joint probability and conditional probability. We discuss also the concept of marginal probability. Further, we describe some theorems of matrices of real elements including quadratic form.