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... 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 natura...

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...division; some basic properties of the set REAL (e.g. density), and the scheme of separation for sets of reals. MML Identifier:REAL_1. WWW:http://mizar.org/JFM/Vol1/real_1.html The articles [2], [4], =-=[1]-=-, and [3] provide the notation and terminology for this paper. In this paper x, y, z, t denote real numbers. Let us note that every element of R is real. A real number is an element of R. Let x be a r...

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...Some theorems about properties of finite sets and finite families of sets are proved. MML Identifier: FINSET_1. WWW: http://mizar.org/JFM/Vol1/finset_1.html The articles [6], [4], [7], [8], [3], [5], =-=[1]-=-, and [2] provide the notation and terminology for this paper. Let I1 be a set. We say that I1 is finite if and only if: (Def. 1) There exists a function p such that rng p = I1 and dom p ∈ ω. We intro...

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...enoted by element of D n . Also some basic facts about tuples of a non-empty set are proved. MML Identifier: FINSEQ_2. WWW: http://mizar.org/JFM/Vol2/finseq_2.html The articles [13], [12], [9], [16], =-=[2]-=-, [3], [14], [11], [1], [15], [17], [6], [8], [7], [4], [5], and [10] provide the notation and terminology for this paper. For simplicity, we adopt the following convention: i, j, l are natural number...

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...those notions are proved. MML Identifier:RLVECT_3. WWW:http://mizar.org/JFM/Vol2/rlvect_3.html The articles [8], [6], [16], [10], [3], [17], [2], [4], [5], [14], [9], [7], [13], [12], [11], [15], and =-=[1]-=- provide the notation and terminology for this paper. For simplicity, we follow the rules: x, X, Y , Z are sets, a, b are real numbers, V is a real linear space, W1, W2, W3 are subspaces of V , v, v1,...

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...element of the range. At the end of the article we prove the pigeon hole principle. MML Identifier: FINSEQ_4. WWW: http://mizar.org/JFM/Vol2/finseq_4.html The articles [7], [10], [8], [3], [11], [4], =-=[1]-=-, [5], [6], [2], and [9] provide the notation and terminology for this paper. For simplicity, we adopt the following rules: f is a function, p, q are finite sequences, A, B, x, y, z are sets, and i, k...

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...x ∈ Y equivalent only for sets that yield non empty values. Therefore the care is advised. MML Identifier:PBOOLE. WWW:http://mizar.org/JFM/Vol5/pboole.html The articles [8], [9], [10], [2], [7], [3], =-=[1]-=-, [5], and [4] provide the notation and terminology for this paper. 1. PRELIMINARIES In this paper i, e are sets. Let f be a function. Let us observe that f is empty yielding if and only if: (Def. 1) ...

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...proved some properties of real numbers and set representations of binary operations on real numbers. MML Identifier: RVSUM_1. WWW: http://mizar.org/JFM/Vol2/rvsum_1.html The articles [17], [21], [8], =-=[2]-=-, [18], [12], [1], [19], [22], [5], [7], [6], [4], [15], [14], [16], [3], [10], [20], [13], and [9] provide the notation and terminology for this paper. For simplicity, we adopt the following rules: i...

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... limit of sequences. In the article there are some properties of real numbers useful in the other theorems of this article. MML Identifier:SEQ_2. WWW:http://mizar.org/JFM/Vol1/seq_2.html The articles =-=[1]-=-, [6], [3], [5], [7], [2], and [4] provide the notation and terminology for this paper. For simplicity, we adopt the following rules: n, m are natural numbers, r, r1, p, g1, g are real numbers, s1, s ...