Summary. The basic purpose of the paper is to prepare preliminaries of the theory of many sorted algebras. The concept of the signature of a many sorted algebra is introduced as well as the concept of many sorted algebra itself. Some auxiliary related notions are defined. The correspondence between (1 sorted) universal algebras [8] and many sorted algebras with one sort only is described by introducing two functors mapping one into the other. The construction is done this way that the composition of both functors is the identity on universal algebras.

Summary. Some operations on the set of n-tuples of real numbers are introduced. Addition, difference of such n-tuples, complement of a n-tuple 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 n-tuples of real numbers and square power of n-tuple 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 n-tuples 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.

Summary. The aim is to prove, using Mizar System, one of the most important result in general topology, namely the Stone Theorem on paracompactness of metrizable spaces [18]. Our proof is based on [17] (and also [15]). We prove first auxiliary fact that every open cover of any metrizable space has a locally finite open refinement. We show next the main theorem that every metrizable space is paracompact. The remaining material is devoted to concepts and certain properties needed for the formulation and the proof of that theorem (see also [4]).

this paper. 1. COMBINING OF MANY SORTED SIGNATURES Let S be a many sorted signature. A gate of S is an element of the operation symbols of S

this paper. In this paper k, n are natural numbers and r is a real number. Let us consider n. The functor R

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Summary. A supplement of [3] and [2], i.e. some useful and explanatory properties of the product and also the curried and uncurried functions are shown. Besides, the functions yielding functions are considered: two different products and other operation of such functions are introduced. Finally, two facts are presented: quasi-distributivity of the power of the set to other one w.r.t. the union (X � x f (x) ≈ ∏x X f (x) ) and quasi-distributivity of the product w.r.t. the raising to the power (∏x f (x) X ≈ (∏x f (x)) X).

Summary. The article contains some propositions and theorems related to [9] and [8]. The notions introduced in [9] are extended to finite sequences. A number of additional propositions related to this notions are proved. There are also proved some properties of distributive operations and unary operations. The notation and propositions for inverses are introduced.

Summary. We define the following functions: exponential function (for natural exponent), factorial function and Newton coefficients. We prove some basic properties of notions introduced. There is also a proof of binominal formula. We prove also that ∑ n �n � k=0 k = 2n.

this article. Operations on