Summary. Sequences in metric spaces are defined. The article contains definitions of bounded, convergent, Cauchy sequences. The subsequences are introduced too. Some theorems concerning sequences are proved.

Summary. A continuation of [6]. It contains the definitions of the convergent sequence

We present a formalization of the factor theorem for univariate polynomials, also called the (little) Bezout theorem: Let r belong to a commutative ring L and p(x) be a polynomial over L. Then x − r divides p(x) iff p(r) = 0. We also prove some consequences of this theorem like that any non zero polynomial of degree n over an algebraically closed integral domain has n (non necessarily distinct) roots.

On this article, the basic properties of linear spaces which are defined by the set of all linear operators from one linear space to another are described. Especially, the Banach space is introduced. This is defined by the set of all bounded linear operators.

Summary. We present a Mizar formalization of chapter 4.4 of [8] devoted to special orderings in additive monoids to be used for ordering terms in multivariate polynomials. We have extended the treatment to the case of infinite number of variables. It turns out that in such case admissible orderings are not necessarily well orderings. MML Identifier:BAGORDER. WWW:http://mizar.org/JFM/Vol14/bagorder.html

Summary. This article contains a definition of three classes of topological spaces: first-countable, Frechet, and sequential. Next there are some facts about them, that every first-countable space is Frechet and every Frechet space is sequential. Next section contains a formalized construction of topological space which is Frechet but not first-countable. This article is based on [10, pp. 73–81].

this paper. 1. Preliminaries

Summary. We present a formalization of roots of unity, define cyclotomic polynomials and demonstrate the relationship between cyclotomic polynomials and unital polynomials.

Let T be a non empty 1-sorted structure, let f be a function from N into N, and let S be a sequence of T. Then S · f is a sequence of T. Next we state two propositions: (1) Let T be a non empty 1-sorted structure, S be a sequence of T, and N1 be an increasing sequence of naturals. Then S · N1 is a sequence of T. (2) For every sequence R1 of real numbers such that R1 = idN holds R1 is an increasing sequence of naturals. Let T be a non empty 1-sorted structure and let S be a sequence of T. A sequence of T is called a subsequence of S if: (Def. 1) There exists an increasing sequence N1 of naturals such that it = S · N1. The following propositions are true: (3) For every non empty 1-sorted structure T holds every sequence S of T is a subsequence of S. (4) For every non empty 1-sorted structure T and for every sequence S of T and for every subsequence S1 of S holds rngS1 ⊆ rngS. Let T be a non empty 1-sorted structure, let N1 be an increasing sequence of naturals, and let S be a sequence of T. Then S · N1 is a subsequence of S. One can prove the following proposition (5) Let T be a non empty 1-sorted structure, S1 be a sequence of T, and S2 be a subsequence of S1. Then every subsequence of S2 is a subsequence of S1. In this article we present several logical schemes. The scheme SubSeqChoice deals with a non empty 1-sorted structureA, a sequenceB ofA, and a unary predicateP, and states that:

this paper. For simplicity, we adopt the following convention: X denotes a real unitary space, a, b, r denote real numbers, s 1 , s 2 , s 3 denote sequences of X , R 1 , R 2 , R 3 denote sequences of real numbers, and k, n, m denote natural numbers. The scheme Rec Func Ex RUS deals with a real unitary space A; a point B of A; and a binary functor F yielding a point of A; and states that: There exists a function f from N into the carrier of A such that f(0) = B and for every element n of N and for every point x of A such that x = f(n) holds