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514
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 53 (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)
Real Function Continuity
, 2002
"... this paper. For simplicity, we adopt the following convention: n denotes a natural number, X , X 1 , Z, Z 1 denote sets, s, g, r, p, x 0 , x 1 , x 2 denote real numbers, s 1 denotes a sequence of real numbers, Y denotes a subset of R, and f , f 1 , f 2 denote partial functions from R to R ..."
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Cited by 49 (8 self)
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this paper. For simplicity, we adopt the following convention: n denotes a natural number, X , X 1 , Z, Z 1 denote sets, s, g, r, p, x 0 , x 1 , x 2 denote real numbers, s 1 denotes a sequence of real numbers, Y denotes a subset of R, and f , f 1 , f 2 denote partial functions from R to R
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 47 (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.
Subspaces and cosets of subspaces in real linear space
 Formalized Mathematics
, 1990
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Topological Spaces
"... this paper. The following propositions are true: (1) Let A, B be non empty sets and R 1 , R 2 be relations between A and B. Suppose that for every element x of A and for every element y of B holds hhx; yii 2 R 1 iff hhx; ..."
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Cited by 34 (0 self)
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this paper. The following propositions are true: (1) Let A, B be non empty sets and R 1 , R 2 be relations between A and B. Suppose that for every element x of A and for every element y of B holds hhx; yii 2 R 1 iff hhx;
The limit of a real function at infinity
 Journal of Formalized Mathematics
, 1990
"... Summary. We introduced the halflines (open and closed), real sequences divergent to infinity (plus and minus) and the proper and improper limit of a real function at infinty. We prove basic properties of halflines, sequences divergent to infinity and the limit of function at infinity. ..."
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Cited by 33 (6 self)
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Summary. We introduced the halflines (open and closed), real sequences divergent to infinity (plus and minus) and the proper and improper limit of a real function at infinty. We prove basic properties of halflines, sequences divergent to infinity and the limit of function at infinity.