Summary. In the paper some notions useful in formalization of [11] are introduced, e.g. the definition of the poset of subsets of a set with inclusion as an ordering relation. Using the theory of many sorted sets authors formulate the definition of product of relational structures.

The paper introduces the concept of a net (a generalized sequence). The goal is to enable the continuation of the translation of [14].

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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.

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this paper. 1. Preliminaries For simplicity, we follow the rules: I, G, H, i are sets, A, B, M are many sorted sets indexed by I, s 1 , s 2 , s 3 are families of subsets of I, v, w are subsets of I, and F is a many sorted function indexed by I

The class of continuous lattices can be characterized by infinitary equations. Therefore, it is closed under the formation of subalgebras and homomorphic images. Following the terminology of [18] we introduce a continuous lattice subframe to be a sublattice closed under the formation of arbitrary infs and directed sups. This notion corresponds with a subalgebra of a continuous lattice in [16]. The class of completely distributive lattices is also introduced in the paper. Such lattices are complete and satisfy the most restrictive type of the general distributivity law. Obviously each completely distributive lattice is a Heyting algebra. It was hard to find the best Mizar implementation of the complete distributivity equational condition (denoted by CD in [16]). The powerful and well developed Many Sorted Theory gives the most convenient way of this formalization. A set double indexed by K, introduced in the paper, corresponds with a family {x j,k: j ∈ J,k ∈ K ( j)}. It is defined to be a suitable many sorted function. Two special functors: Sups and Infs as counterparts of Sup and Inf respectively, introduced in [33], are also defined. Originally the equation in Definition 2.4 of [16, p. 58] looks as follows: j∈J k∈K ( j)x j,k = � � f ∈M j∈Jx j, f ( j), where M is the set of functions defined on J with values f(j) ∈ K ( j).

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