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The papers [18], [9], [21], [14], [23], [6], [24], [2], [3], [8], [10], [1], [22], [7], [11], [20], [16], [19], [4], [5], [13], [12], [17], and [15] provide the terminology and notation for this paper. For simplicity, we adopt the following convention: k, l, m, n, i, j denote natural numbers, K, N denote non empty subsets of N, K1, N1, M1 denote subsets of(More)
The existing examples of natural deduction proofs, either declarative or procedural, indicate that often the legibility of proof scripts is of secondary importance to the authors. As soon as the computer accepts the proof script, many authors do not work on improving the parts that could be shortened and do not avoid repetitions of technical sub-deductions,(More)
  • Karol Pak
  • Logical Methods in Computer Science
  • 2014
In formal proof checking environments such as Mizar it is not merely the validity of mathematical formulas that is evaluated in the process of adoption to the body of accepted formalizations, but also the readability of the proofs that witness validity. As in case of computer programs, such proof scripts may sometimes be more and sometimes be less readable.(More)
In this paper I present selected properties of triangular matrices and basic properties of the rank of matrices over a field. I define a submatrix as a matrix formed by selecting certain rows and columns from a bigger matrix. That is in my considerations, as an array, it is cut down to those entries constrained by row and column. Then I introduce the(More)
We continue Mizar formalization of general topology according to the book [16] by Engelking. In the article, we present the final theorem of Section 4.1. Namely, the paper includes the formalization of theorems on the correspondence between the cardinalities of the basis and of some open subcover, and a discreet (closed) subspaces, and the weight of that(More)
The articles [21], [10], [24], [17], [26], [6], [27], [2], [9], [11], [1], [25], [7], [8], [22], [19], [5], [15], [12], [20], [16], [14], [18], [13], [3], [23], and [4] provide the terminology and notation for this paper. For simplicity, we use the following convention: x, x1, x2, y, z, X ′ denote sets, X, Y denote finite sets, n, k, m denote natural(More)
In this article we describe the notion of affinely independent subset of a real linear space. First we prove selected theorems concerning operations on linear combinations. Then we introduce affine independence and prove the equivalence of various definitions of this notion. We also introduce the notion of the affine hull, i.e. a subset generated by a set(More)
For simplicity, we adopt the following rules: T , T1, T2 denote topological spaces, A, B denote subsets of T , F denotes a subset of T A, G, G1, G2 denote families of subsets of T , U , W denote open subsets of T A, p denotes a point of T A, n denotes a natural number, and I denotes an integer. One can prove the following propositions: (1) Fr(B ∩A) ⊆ FrB(More)