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284
Group and field definitions
 Journal of Formalized Mathematics
, 1989
"... Summary. The article contains exactly the same definitions of group and field as those in [4]. These definitions were prepared without the help of the definitions and properties of Nat and Real modes included in the MML. This is the first of a series of articles in which we are going to introduce th ..."
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Cited by 97 (1 self)
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Summary. The article contains exactly the same definitions of group and field as those in [4]. These definitions were prepared without the help of the definitions and properties of Nat and Real modes included in the MML. This is the first of a series of articles in which we are going to introduce the concept of the set of real numbers in a elementary axiomatic way.
Construction of rings and left, right, and bimodules over a ring
 Journal of Formalized Mathematics
, 1990
"... Summary. Definitions of some classes of rings and left, right, and bimodules over a ring and some elementary theorems on rings and skew fields. ..."
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Cited by 63 (16 self)
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Summary. Definitions of some classes of rings and left, right, and bimodules over a ring and some elementary theorems on rings and skew fields.
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.
Categories of Groups
, 2000
"... this paper. In this paper x, y denote sets, D denotes a non empty set, and U 1 denotes a universal class. The following propositions are true: (2) ..."
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Cited by 39 (4 self)
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this paper. In this paper x, y denote sets, D denotes a non empty set, and U 1 denotes a universal class. The following propositions are true: (2)
Subspaces and cosets of subspaces in real linear space.
 Formalized Mathematics,
, 1990
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Subspaces and cosets of subspaces in vector space.
 Formalized Mathematics,
, 1990
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Transpose Matrices and Groups of Permutations
, 2003
"... Some facts concerning matrices with dimension 2 × 2 are shown. Upper and lower triangular matrices, and operation of deleting rows and columns in a matrix are introduced. Besides, we deal with sets of permutations and the fact that all permutations of finite set constitute a finite group is proved. ..."
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Cited by 34 (0 self)
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Some facts concerning matrices with dimension 2 × 2 are shown. Upper and lower triangular matrices, and operation of deleting rows and columns in a matrix are introduced. Besides, we deal with sets of permutations and the fact that all permutations of finite set constitute a finite group is proved. Some proofs are based on [11] and [14].
Homomorphisms and isomorphisms of groups. Quotient group
 Journal of Formalized Mathematics
, 1991
"... Summary. Quotient group, homomorphisms and isomorphisms of groups are introduced. The so called isomorphism theorems are proved following [9]. ..."
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Cited by 34 (0 self)
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Summary. Quotient group, homomorphisms and isomorphisms of groups are introduced. The so called isomorphism theorems are proved following [9].
Linear combinations in real linear space
 Journal of Formalized Mathematics
, 1990
"... Summary. The article is continuation of [17]. At the beginning we prove some theorems concerning sums of finite sequence of vectors. We introduce the following notions: sum of finite subset of vectors, linear combination, carrier of linear combination, linear combination of elements of a given set o ..."
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Cited by 34 (5 self)
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Summary. The article is continuation of [17]. At the beginning we prove some theorems concerning sums of finite sequence of vectors. We introduce the following notions: sum of finite subset of vectors, linear combination, carrier of linear combination, linear combination of elements of a given set of vectors, sum of linear combination. We also show that the set of linear combinations is a real linear space. At the end of article we prove some auxiliary theorems that should be proved in [8], [5], [9], [2] or [10].
Real normed space
 Formalized Mathematics
, 1991
"... Summary. We construct a real normed space 〈V, �.�〉, where V is a real vector space and �. � is a norm. Auxillary properties of the norm are proved. Next, we introduce a notion of sequence in the real normed space. The basic operations on sequences (addition, subtraction, multiplication by real numbe ..."
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Cited by 32 (0 self)
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Summary. We construct a real normed space 〈V, �.�〉, where V is a real vector space and �. � is a norm. Auxillary properties of the norm are proved. Next, we introduce a notion of sequence in the real normed space. The basic operations on sequences (addition, subtraction, multiplication by real number) are defined. We study some properties of sequences in the real normed space and the operations on them.