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687
Convergent Real Sequences. Upper and Lower Bound of Sets of Real Numbers
, 2000
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Factorial and Newton coefficients
 Journal of Formalized Mathematics
, 1990
"... Summary. We define the following functions: exponential function (for natural exponent), factorial function and Newton coefficients. We prove some basic properties of notions introduced. There is also a proof of binominal formula. We prove also that ∑ n �n � k=0 k = 2n. ..."
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Cited by 85 (0 self)
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Summary. We define the following functions: exponential function (for natural exponent), factorial function and Newton coefficients. We prove some basic properties of notions introduced. There is also a proof of binominal formula. We prove also that ∑ n �n � k=0 k = 2n.
The Euclidean Space
, 1991
"... this paper. In this paper k, n are natural numbers and r is a real number. Let us consider n. The functor R ..."
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Cited by 83 (0 self)
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this paper. In this paper k, n are natural numbers and r is a real number. Let us consider n. The functor R
Families of Subsets, Subspaces and Mappings in Topological Spaces
"... This article is a continuation of [13]. Some basic theorems about families of sets in a topological space have been proved. Following redefinitions have been made: singleton of a set as a family in the topological space and results of boolean operations on families as a family of the topological spa ..."
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Cited by 62 (0 self)
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This article is a continuation of [13]. Some basic theorems about families of sets in a topological space have been proved. Following redefinitions have been made: singleton of a set as a family in the topological space and results of boolean operations on families as a family of the topological space. Notion of a family of complements of sets and a closed (open) family have been also introduced. Next some theorems refer to subspaces in a topological space: some facts about types in a subspace, theorems about open and closed sets and families in a subspace. A notion of restriction of a family has been also introduced and basic properties of this notion have been proved. The last part of the article is about mappings. There are proved necessary and sufficient conditions for a mapping to be continuous. A notion of homeomorphism has been defined next. Theorems about homeomorphisms of topological spaces have been also proved. MML Identifier: TOPS2.
Joining of decorated trees
 Journal of Formalized Mathematics
, 1993
"... Summary. This is the continuation of the sequence of articles on trees (see [2], [4], [5]). The main goal is to introduce joining operations on decorated trees corresponding with operations introduced in [5]. We will also introduce the operation of substitution. In the last section we dealt with tre ..."
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Cited by 56 (19 self)
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Summary. This is the continuation of the sequence of articles on trees (see [2], [4], [5]). The main goal is to introduce joining operations on decorated trees corresponding with operations introduced in [5]. We will also introduce the operation of substitution. In the last section we dealt with trees decorated by Cartesian product, i.e. we showed some lemmas on joining operations applied to such trees.
On the Decomposition of the States of SCM
, 1993
"... This article continues the development of the basic terminology ..."
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Cited by 52 (1 self)
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This article continues the development of the basic terminology
Subgroup and cosets of subgroups
 Journal of Formalized Mathematics
, 1990
"... Summary. We introduce notion of subgroup, coset of a subgroup, sets of left and right cosets of a subgroup. We define multiplication of two subset of a group, subset of reverse elemens of a group, intersection of two subgroups. We define the notion of an index of a subgroup and prove Lagrange theore ..."
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Cited by 50 (9 self)
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Summary. We introduce notion of subgroup, coset of a subgroup, sets of left and right cosets of a subgroup. We define multiplication of two subset of a group, subset of reverse elemens of a group, intersection of two subgroups. We define the notion of an index of a subgroup and prove Lagrange theorem which states that in a finite group the order of the group equals the order of a subgroup multiplied by the index of the subgroup. Some theorems that belong rather to [1] are proved.
On Defining Functions on Trees
, 2003
"... this paper. 1. PRELIMINARIES One can prove the following propositions: (1) For every non empty set D holds every finite sequence of elements of FinTrees(D) is a finite sequence of elements of Trees(D) ..."
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Cited by 50 (26 self)
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this paper. 1. PRELIMINARIES One can prove the following propositions: (1) For every non empty set D holds every finite sequence of elements of FinTrees(D) is a finite sequence of elements of Trees(D)