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66
The sum and product of finite sequences of real numbers
 Journal of Formalized Mathematics
, 1990
"... Summary. Some operations on the set of ntuples of real numbers are introduced. Addition, difference of such ntuples, complement of a ntuple and multiplication of these by real numbers are defined. In these definitions more general properties of binary operations applied to finite sequences from [ ..."
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Cited by 118 (2 self)
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Summary. Some operations on the set of ntuples of real numbers are introduced. Addition, difference of such ntuples, complement of a ntuple and multiplication of these by real numbers are defined. In these definitions more general properties of binary operations applied to finite sequences from [9] are used. Then the fact that certain properties are satisfied by those operations is demonstrated directly from [9]. Moreover some properties can be recognized as being those of real vector space. Multiplication of ntuples of real numbers and square power of ntuple of real numbers using for notation of some properties of finite sums and products of real numbers are defined, followed by definitions of the finite sum and product of ntuples of real numbers using notions and properties introduced in [11]. A number of propositions and theorems on sum and product of finite sequences of real numbers are proved. As additional properties there are proved some properties of real numbers and set representations of binary operations on real numbers.
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 92 (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 45 (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 vector space
 Journal of Formalized Mathematics
, 1990
"... Summary. We introduce the notions of subspace of vector space and coset of a subspace. We prove a number of theorems concerning those notions. Some theorems that belong rather to [4] are proved. ..."
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Cited by 34 (6 self)
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Summary. We introduce the notions of subspace of vector space and coset of a subspace. We prove a number of theorems concerning those notions. Some theorems that belong rather to [4] are proved.
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 28 (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].
Lattice of subgroups of a group. Frattini subgroup
 Journal of Formalized Mathematics
, 1990
"... Summary. We define the notion of a subgroup generated by a set of element of a group and two closely connected notions. Namely lattice of subgroups and Frattini subgroup. The operations in the lattice are the intersection of subgroups (introduced in [21]) and multiplication of subgroups which result ..."
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Cited by 28 (5 self)
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Summary. We define the notion of a subgroup generated by a set of element of a group and two closely connected notions. Namely lattice of subgroups and Frattini subgroup. The operations in the lattice are the intersection of subgroups (introduced in [21]) and multiplication of subgroups which result is defined as a subgroup generated by a sum of carriers of the two subgroups. In order to define Frattini subgroup and to prove theorems concerning it we introduce notion of maximal subgroup and nongenerating element of the group (see [9, page 30]). Frattini subgroup is defined as in [9] as an intersection of all maximal subgroups. We show that an element of the group belongs to Frattini subgroup of the group if and only if it is a nongenerating element. We also prove theorems that should be proved in [1] but are not.
The field of complex numbers
 Journal of Formalized Mathematics
"... [14], [5], [10], and [9] provide the notation and terminology for this paper. The following propositions are true: 1. PRELIMINARIES (1) For every element z of C holds z  = z. (2) For all real numbers x1, y1, x2, y2 holds (x1 + y1i) ·(x2 + y2i) = (x1 · x2 − y1 · y2)+(x1 · y2 + x2 · y1)i. (3) ..."
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Cited by 26 (1 self)
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[14], [5], [10], and [9] provide the notation and terminology for this paper. The following propositions are true: 1. PRELIMINARIES (1) For every element z of C holds z  = z. (2) For all real numbers x1, y1, x2, y2 holds (x1 + y1i) ·(x2 + y2i) = (x1 · x2 − y1 · y2)+(x1 · y2 + x2 · y1)i. (3) For every real number r holds (r+0i) · i = 0+ri. (4) For every real number r holds r+0i  = r. (5) For every element z of C such that z  � = 0 holds z+0i = z z+0i · z. 2. SOME FACTS ON THE FIELD OF COMPLEX NUMBERS Let x, y be real numbers. The functor x+yiCF yielding an element of CF is defined as follows: (Def. 1) x+yiCF = x+yi. The element iCF of CF is defined as follows: (Def. 2) iCF = i. We now state several propositions: (6) iCF = 0+1i and iCF = 0+1iCF.
Classes of conjugation. Normal subgroups
 Journal of Formalized Mathematics
, 1990
"... article we define notion of conjugation for elements, subsets and subgroups of a group. We define the classes of conjugation. Normal subgroups of a group and normalizator of a subset of a group or of a subgroup are introduced. We also define the set of all subgroups of a group. An auxiliary theorem ..."
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Cited by 26 (7 self)
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article we define notion of conjugation for elements, subsets and subgroups of a group. We define the classes of conjugation. Normal subgroups of a group and normalizator of a subset of a group or of a subgroup are introduced. We also define the set of all subgroups of a group. An auxiliary theorem that belongs rather to [2] is proved.