function from a set X into a set Y, denoted by “Function of X,Y ”, the set of all functions from a set X into a set Y, denoted by Funcs(X,Y), and the permutation of a set (mode Permutation of X, where X is a set). Theorems and schemes included in the article are reformulations of the theorems of [1] in the new terminology. Also some basic facts about functions of two variables are proved.

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.

Summary. We present the choice function rule in the beginning of the article. In the main part of the article we formalize the base of cardinal theory. In the first section we introduce the concept of cardinal numbers and order relations between them. We present here Cantor-Bernstein theorem and other properties of order relation of cardinals. In the second section we show that every set has cardinal number equipotence to it. We introduce notion of alephs and we deal with the concept of finite set. At the end of the article we show two schemes of cardinal induction. Some definitions are based on [9] and [10].

this paper. A real number is an element of R

this article we prove some auxiliary theorems and schemes related to the articles: [1] and [2]. MML Identifier: PARTFUN1. WWW: http://mizar.org/JFM/Vol1/partfun1.html The articles [4], [6], [3], [5], [7], [8], and [1] provide the notation and terminology for this paper. We adopt the following rules: x, y, y 1 , y 2 , z, z 1 , z 2 denote sets, P , Q, X , X 0 , X 1 , X 2 , Y , Y 0 , Y 1 , Y 2 , V , Z denote sets, and C, D denote non empty sets. We now state three propositions: (1) If P ` [: X 1

Summary. In this paper we define binary and unary operations on domains. We also define the following predicates concerning the operations:... is commutative,... is associative,... is the unity of..., and... is distributive wrt.... A number of schemes useful in justifying the existence of the operations are proved. MML Identifier:BINOP_1. WWW:http://mizar.org/JFM/Vol1/binop_1.html The articles [4], [3], [5], [6], [1], and [2] provide the notation and terminology for this paper. Let f be a function and let a, b be sets. The functor f(a, b) yielding a set is defined by: (Def. 1) f(a, b) = f(〈a, b〉). In the sequel A is a set. Let A, B be non empty sets, let C be a set, let f be a function from [:A, B:] into C, let a be an element of A, and let b be an element of B. Then f(a, b) is an element of C. The following proposition is true (2) 1 Let A, B, C be non empty sets and f1, f2 be functions from [:A, B:] into C. Suppose that for every element a of A and for every element b of B holds f1(a, b) = f2(a, b). Then f1 = f2. Let A be a set. A unary operation on A is a function from A into A. A binary operation on A is a

Summary. The article contains the definition of a finite set based on the notion of finite sequence. Some theorems about properties of finite sets and finite families of sets are proved.

Summary. The article contains definitions of the following concepts: family of sets, family of subsets of a set, the intersection of a family of sets. Functors ∪, ∩, and \ are redefined for families of subsets of a set. Some properties of these notions are presented.

Summary. The article includes: theorems related to domains, theorems related to Cartesian products presented earlier in various articles and simplified here by substituting domains for sets and omitting the assumption that the sets involved must not be empty. Several schemes and theorems related to Fraenkel operator are given. We also redefine subset yielding functions such as the pair of elements of a set and the union of two subsets of a set.

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 non-empty set D and it is denoted by element of D