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The fundamental properties of natural numbers
 Journal of Formalized Mathematics
, 1989
"... Summary. Some fundamental properties of addition, multiplication, order relations, exact division, the remainder, divisibility, the least common multiple, the greatest common divisor are presented. A proof of Euclid algorithm is also given. MML Identifier:NAT_1. WWW:http://mizar.org/JFM/Vol1/nat_1.h ..."
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Summary. Some fundamental properties of addition, multiplication, order relations, exact division, the remainder, divisibility, the least common multiple, the greatest common divisor are presented. A proof of Euclid algorithm is also given. MML Identifier:NAT_1. WWW:http://mizar.org/JFM/Vol1/nat_1.html The articles [4], [6], [1], [2], [5], and [3] provide the notation and terminology for this paper. A natural number is an element of N. For simplicity, we use the following convention: x is a real number, k, l, m, n are natural numbers, h, i, j are natural numbers, and X is a subset of R. The following proposition is true (2) 1 For every X such that 0 ∈ X and for every x such that x ∈ X holds x+1 ∈ X and for every k holds k ∈ X. Let n, k be natural numbers. Then n+k is a natural number. Let n, k be natural numbers. Note that n+k is natural. In this article we present several logical schemes. The scheme Ind concerns a unary predicate P, and states that: For every natural number k holdsP[k] provided the parameters satisfy the following conditions: • P[0], and • For every natural number k such thatP[k] holdsP[k+1]. The scheme Nat Ind concerns a unary predicateP, and states that: For every natural number k holdsP[k] provided the following conditions are satisfied: • P[0], and • For every natural number k such thatP[k] holdsP[k+1]. Let n, k be natural numbers. Then n · k is a natural number. Let n, k be natural numbers. Observe that n · k is natural. Next we state several propositions: (18) 2 0 ≤ i. (19) If 0 � = i, then 0 < i. (20) If i ≤ j, then i · h ≤ j · h. 1 The proposition (1) has been removed. 2 The propositions (3)–(17) have been removed.
Finite Sequences and Tuples of Elements of a Nonempty Sets
, 1990
"... this article is the definition of tuples. The element of a set of all sequences of the length n of D is called a tuple of a nonempty set D and it is denoted by element of D ..."
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Cited by 332 (7 self)
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this article is the definition of tuples. The element of a set of all sequences of the length n of D is called a tuple of a nonempty set D and it is denoted by element of D
Basis of Real Linear Space
, 1990
"... this paper. For simplicity, we follow the rules: x is a set, a, b are real numbers, V is a real linear space, W 1 , W 2 , W 3 are subspaces of V , v, v 1 , v 2 are vectors of V , A, B are subsets of the carrier of V , L, L 1 , L 2 are linear combinations of V , l is a linear combination of A, F , G ..."
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Cited by 285 (21 self)
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this paper. For simplicity, we follow the rules: x is a set, a, b are real numbers, V is a real linear space, W 1 , W 2 , W 3 are subspaces of V , v, v 1 , v 2 are vectors of V , A, B are subsets of the carrier of V , L, L 1 , L 2 are linear combinations of V , l is a linear combination of A, F , G
Topological Properties of Subsets in Real Numbers
 JOURNAL OF FORMALIZED MATHEMATICS
, 2002
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The sum and product of finite sequences of real numbers.
 Formalized Mathematics,
, 1990
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Convergent sequences and the limit of sequences.
 Formalized Mathematics,
, 1990
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The complex numbers
 Journal of Formalized Mathematics
, 1990
"... Summary. We define the set C of complex numbers as the set of all ordered pairs z = 〈a,b 〉 where a and b are real numbers and where addition and multiplication are defined. We define the real and imaginary parts of z and denote this by a = ℜ(z), b = ℑ(z). These definitions satisfy all the axioms for ..."
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Cited by 118 (1 self)
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Summary. We define the set C of complex numbers as the set of all ordered pairs z = 〈a,b 〉 where a and b are real numbers and where addition and multiplication are defined. We define the real and imaginary parts of z and denote this by a = ℜ(z), b = ℑ(z). These definitions satisfy all the axioms for a field. 0C = 0 + 0i and 1C = 1 + 0i are identities for addition and multiplication respectively, and there are multiplicative inverses for each non zero element in C. The difference and division of complex numbers are also defined. We do not interpret the set of all real numbers R as a subset of C. From here on we do not abandon the ordered pair notation for complex numbers. For example: i 2 = (0+1i) 2 = −1+0i � = −1. We conclude this article by introducing two operations on C which are not field operations. We define the absolute value of z denoted by z  and the conjugate of z denoted by z ∗.
Monotone Real Sequences. Subsequences
"... this paper. We follow the rules: n, m, k are natural numbers, r is a real number, and s 1 , s 2 , s 3 are sequences of real numbers. Let s 1 be a partial function from N to R. We say that s 1 is increasing if and only if: (Def. 1) For every n holds s 1 (n) ! s 1 (n + 1): We say that s 1 is decreasin ..."
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Cited by 96 (9 self)
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this paper. We follow the rules: n, m, k are natural numbers, r is a real number, and s 1 , s 2 , s 3 are sequences of real numbers. Let s 1 be a partial function from N to R. We say that s 1 is increasing if and only if: (Def. 1) For every n holds s 1 (n) ! s 1 (n + 1): We say that s 1 is decreasing if and only if: (Def. 2) For every n holds s 1 (n + 1) ! s 1 (n): We say that s 1 is nondecreasing if and only if: (Def. 3) For every n holds s 1 (n) s 1 (n + 1): We say that s 1 is nonincreasing if and only if: (Def. 4) For every n holds s 1 (n + 1) s 1 (n): Let f be a function. We say that f is constant if and only if: (Def. 5) For all sets n 1 , n 2 such that n 1 2 dom f and n 2 2 domf holds f(n 1 ) = f(n 2 ): Let us consider s 1 . Let us observe that s 1 is constant if and only if: (Def. 6) There exists r such that for every n holds s 1 (n) = r:
An Overview of the MIZAR Project
 UNIVERSITY OF TECHNOLOGY, BASTAD
, 1992
"... The Mizar project is a longterm effort aimed at developing software to support a working mathematician in preparing papers. A. Trybulec, the leader of the project, has designed a language for writing formal mathematics. The logical structure of the language is based on a natural deduction system ..."
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Cited by 94 (1 self)
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The Mizar project is a longterm effort aimed at developing software to support a working mathematician in preparing papers. A. Trybulec, the leader of the project, has designed a language for writing formal mathematics. The logical structure of the language is based on a natural deduction system developed by Ja'skowski. The texts written in the language are called Mizar articles and are organized into a data base. The TarskiGrothendieck set theory forms the basis of doing mathematics in Mizar. The implemented processor of the language checks the articles for logical consistency and correctness of references to other articles.