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

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

Summary. We introduce the notion of a predicate that states that a function is one-toone at a given element of its domain (i.e. counterimage of image of the element is equal to its singleton). We also introduce some rather technical functors concerning finite sequences: the lowest index of the given element of the range of the finite sequence, the substring preceding (and succeeding) the first occurrence of given element of the range. At the end of the article we prove the pigeon hole principle.

Summary. In the article we introduce functors yielding to a binary operation its composition with an arbitrary functions on its left side, its right side or both. We prove theorems describing the basic properties of these functors. We introduce also constant functions and converse of a function. The recent concept is defined for an arbitrary function, however is meaningful in the case of functions which range is a subset of a Cartesian product of two sets. Then the converse of a function has the same domain as the function itself and assigns to an element of the domain the mirror image of the ordered pair assigned by the function. In the case of functions defined on a non-empty set we redefine the above mentioned functors and prove simplified versions of theorems proved in the general case. We prove also theorems stating relationships between introduced concepts and such properties of binary operations as commutativity or associativity.

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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this paper (and in another Mizar articles) we work in Tarski-Grothendieck (TG) theory (see [17]) which ensures the existence of sets that have properties like universal class (i.e. this theory is stronger than MK). The sets are introduced in [15] and some concepts of MK are modeled. The concepts are: the class On of all ordinal numbers belonging to the universe, subclasses, transfinite sequences of non-empty elements of universe, etc. The reflection theorem states that if A ¸ is an increasing and continuous transfinite sequence of non-empty sets and class A =

this article we define the choice function of a non-empty set family that does not contain ; as introduced in [6, pages 88--89]. We define order of a set as a relation being reflexive, antisymmetric and transitive in the set, partially ordered set as structure non-empty set and order of the set, chains, lower and upper cone of a subset, initial segments of element and subset of partially ordered set. Some theorems that belong rather to [5] or [14] are proved. MML Identifier: ORDERS1.

this paper. In this paper i, j, k are natural numbers. Let x, y, a, b be sets. The functor (x = y a,b) yields a set and is defined as follows: (Def. 1) (x = y a,b) = a, if x = y, b, otherwise. Let D be a non empty set, let x, y be sets, and let a, b be elements of D. Then (x = y a,b) is an element of D