###### Cartesian Products of Family of Real Linear Spaces

Padlewska and Agata Darmochwał. Topological spaces and continuous functions. Formalized Mathematics , 1( 1 ):223-230, 1990. [14] Jan Popiołek. Real normed space. Formalized Mathematics , 2( 1 ):111-115, 1991. [15] Yasunari Shidama. Banach space of bounded linear operators. Formalized Mathematics , 12( 1 ):39-48, 2004. [16] Andrzej Trybulec. Domains and their Cartesian products. Formalized Mathematics , 1( 1 ):115-122, 1990. [17] Wojciech A. Trybulec. Pigeon hole

###### Topological Manifolds

] Agata Darmochwał. Finite sets. Formalized Mathematics , 1(1):165–167, 1990. [13] Agata Darmochwał. Families of subsets, subspaces and mappings in topological spaces. Formalized Mathematics , 1(2):257–261, 1990. [14] Ryszard Engelking. Teoria wymiaru . PWN, 1981. [15] Adam Grabowski. Properties of the product of compact topological spaces. Formalized Mathematics , 8(1):55–59, 1999. [16] Zbigniew Karno. Separated and weakly separated subspaces of topological spaces. Formalized Mathematics , 2(5):665–674, 1991. [17] Artur Korniłowicz. Jordan curve theorem

###### Products in Categories without Uniqueness of cod and dom

## Summary

The paper introduces Cartesian products in categories without uniqueness of **cod **and **dom**. It is proven that set-theoretical product is the product in the category Ens [7].

###### Probability Measure on Discrete Spaces and Algebra of Real-Valued Random Variables

-65, 1990. [5] Czesław Byliński. Functions from a set to a set. Formalized Mathematics , 1( 1 ):153-164, 1990. [6] Czesław Byliński. Partial functions. Formalized Mathematics , 1( 2 ):357-367, 1990. [7] Czesław Byliński. Some basic properties of sets. Formalized Mathematics , 1( 1 ):47-53, 1990. [8] Czesław Byliński. The sum and product of finite sequences of real numbers. Formalized Mathematics , 1( 4 ):661-668, 1990. [9] Agata Darmochwał

###### Double Sequences and Iterated Limits in Regular Space

## Abstract

First, we define in Mizar [5], the Cartesian product of two filters bases and the Cartesian product of two filters. After comparing the product of two Fréchet filters on ℕ (F_{1}) with the Fréchet filter on ℕ × ℕ (F_{2}), we compare lim_{F₁} and lim_{F₂} for all double sequences in a non empty topological space.

Endou, Okazaki and Shidama formalized in [14] the “convergence in Pringsheim’s sense” for double sequence of real numbers. We show some basic correspondences between the p-convergence and the filter convergence in a topological space. Then we formalize that the double sequence converges in “Pringsheim’s sense” but not in Frechet filter on ℕ × ℕ sense.

In the next section, we generalize some definitions: “is convergent in the first coordinate”, “is convergent in the second coordinate”, “the lim in the first coordinate of”, “the lim in the second coordinate of” according to [14], in Hausdorff space.

Finally, we generalize two theorems: (3) and (4) from [14] in the case of double sequences and we formalize the “iterated limit” theorem (“Double limit” [7], p. 81, par. 8.5 “Double limite” [6] (TG I,57)), all in regular space. We were inspired by the exercises (2.11.4), (2.17.5) [17] and the corrections B.10 [18].

###### Basic Operations on Preordered Coherent Spaces

, 6(1):117-121, 1997. [9] Zbigniew Karno. Separated and weakly separated subspaces of topological spaces. Formalized Mathematics , 2(5):665-674, 1991. [10] Artur Korniłowicz. Cartesian products of relations and relational structures. Formalized Mathematics , 6(1):145-152, 1997. [11] J.L. Krivine. Lambda-calculus, types and models. Ellis & Horwood, 1993. [12] Beata Madras. Product of family of universal algebras. Formalized Mathematics , 4(1):103-108, 1993

###### Definition of Flat Poset and Existence Theorems for Recursive Call

chain-complete posets. Formalized Mathematics, 18(1):47-51, 2010. doi:10.2478/v10037-010-0006-x. [16] Artur Korniłowicz. Cartesian products of relations and relational structures. Formalized Mathematics, 6(1):145-152, 1997. [17] Andrzej Trybulec. Domains and their Cartesian products. Formalized Mathematics, 1(1): 115-122, 1990. [18] Andrzej Trybulec. Tuples, projections and Cartesian products. Formalized Mathematics, 1(1):97-105, 1990. [19] Wojciech A. Trybulec and Grzegorz Bancerek. Kuratowski - Zorn

###### Arithmetic Operations on Functions from Sets into Functional Sets

properties of rational numbers. Formalized Mathematics , 1(5):841-845, 1990. [6] Andrzej Trybulec. Tuples, projections and Cartesian products. Formalized Mathematics , 1(1):97-105, 1990. [7] Andrzej Trybulec. On the sets inhabited by numbers. Formalized Mathematics , 11(4):341-347, 2003. [8] Michał J. Trybulec. Integers. Formalized Mathematics , 1(3):501-505, 1990. [9] Zinaida Trybulec. Properties of subsets. Formalized Mathematics , 1(1):67-71, 1990

###### Z-modules

-367, 1990. Czesław Byliński. Some basic properties of sets. Formalized Mathematics , 1( 1 ):47-53, 1990. Daniele Micciancio and Shafi Goldwasser. Complexity of lattice problems: A cryptographic perspective (the international series in engineering and computer science). 2002. Christoph Schwarzweller. The binomial theorem for algebraic structures. Formalized Mathematics , 9( 3 ):559-564, 2001. Andrzej Trybulec. Domains and their Cartesian products. Formalized

###### Equivalence of Deterministic and Nondeterministic Epsilon Automata

(1):165-167, 1990. [10] Karol Pąk. The Catalan numbers. Part II. Formalized Mathematics , 14(4):153-159, 2006, doi:10.2478/v10037-006-0019-7. [11] Andrzej Trybulec. Domains and their Cartesian products. Formalized Mathematics , 1(1):115-122, 1990. [12] Andrzej Trybulec. Tuples, projections and Cartesian products. Formalized Mathematics , 1(1):97-105, 1990. [13] Michał Trybulec. Formal languages - concatenation and closure. Formalized Mathematics , 15(1):11-15, 2007, doi:10