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Algebraic Approach to Algorithmic Logic

(1): 67-74, 1996. [28] Krzysztof Retel. Properties of first and second order cutting of binary relations. Formalized Mathematics, 13(3):361-365, 2005. [29] Andrzej Trybulec. Binary operations applied to functions. Formalized Mathematics, 1 (2):329-334, 1990. [30] Andrzej Trybulec. Tuples, projections and Cartesian products. Formalized Mathematics, 1(1):97-105, 1990. [31] Andrzej Trybulec. Many sorted algebras. Formalized Mathematics, 5(1):37-42, 1996. [32] Andrzej Trybulec. Many sorted sets

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Tietze Extension Theorem for n-dimensional Spaces

References [1] Grzegorz Bancerek. Cardinal numbers. Formalized Mathematics, 1(2):377-382, 1990. [2] Grzegorz Bancerek. König’s theorem. Formalized Mathematics, 1(3):589-593, 1990. [3] Grzegorz Bancerek. Cartesian product of functions. Formalized Mathematics, 2(4):547-552, 1991. [4] Grzegorz Bancerek. The fundamental properties of natural numbers. Formalized Mathematics, 1(1):41-46, 1990. [5] Grzegorz Bancerek. The ordinal numbers. Formalized Mathematics, 1(1):91-96, 1990

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The Perfect Number Theorem and Wilson's Theorem

Mathematics , 12(1):49-58, 2004. [30] Piotr Rudnicki and Andrzej Trybulec. Abian's fixed point theorem. Formalized Mathematics , 6(3):335-338, 1997. [31] Piotr Rudnicki and Andrzej Trybulec. Multivariate polynomials with arbitrary number of variables. Formalized Mathematics , 9(1):95-110, 2001. [32] Andrzej Trybulec. Binary operations applied to functions. Formalized Mathematics , 1(2):329-334, 1990. [33] Andrzej Trybulec. Tuples, projections and Cartesian products

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Free Term Algebras

References [1] Grzegorz Bancerek. Cardinal numbers. Formalized Mathematics , 1( 2 ):377-382, 1990. [2] Grzegorz Bancerek. Introduction to trees. Formalized Mathematics , 1( 2 ):421-427, 1990. [3] Grzegorz Bancerek. K¨onig’s theorem. Formalized Mathematics , 1( 3 ):589-593, 1990. [4] Grzegorz Bancerek. The ordinal numbers. Formalized Mathematics , 1( 1 ):91-96, 1990. [5] Grzegorz Bancerek. Cartesian product of functions. Formalized Mathematics , 2( 4 ):547-552, 1991

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Program Algebra over an Algebra

References [1] Grzegorz Bancerek. Cardinal numbers. Formalized Mathematics , 1( 2 ):377-382, 1990. [2] Grzegorz Bancerek. Introduction to trees. Formalized Mathematics , 1( 2 ):421-427, 1990. [3] Grzegorz Bancerek. K¨onig’s theorem. Formalized Mathematics , 1( 3 ):589-593, 1990. [4] Grzegorz Bancerek. The ordinal numbers. Formalized Mathematics , 1( 1 ):91-96, 1990. [5] Grzegorz Bancerek. Cartesian product of functions. Formalized Mathematics , 2( 4 ):547-552, 1991

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Characteristic of Rings. Prime Fields

–122, 2008. doi:10.2478/v10037-008-0017-z. [32] Andrzej Trybulec. Domains and their Cartesian products. Formalized Mathematics , 1( 1 ): 115–122, 1990. [33] Andrzej Trybulec. On the sets inhabited by numbers. Formalized Mathematics , 11( 4 ): 341–347, 2003. [34] Michał J. Trybulec. Integers. Formalized Mathematics , 1( 3 ):501–505, 1990. [35] Wojciech A. Trybulec. Groups. Formalized Mathematics , 1( 5 ):821–827, 1990. [36] Wojciech A. Trybulec. Vectors in real linear space. Formalized Mathematics , 1( 2 ):291–296, 1990. [37

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The First Isomorphism Theorem and Other Properties of Rings

Suzuki, and Noboru Endou. Banach algebra of bounded functionals. Formalized Mathematics, 16(2):115-122, 2008. doi:10.2478/v10037-008-0017- z. [32] Andrzej Trybulec. Domains and their Cartesian products. Formalized Mathematics, 1(1): 115-122, 1990. [33] Andrzej Trybulec. Binary operations applied to functions. Formalized Mathematics, 1 (2):329-334, 1990. [34] Andrzej Trybulec. On the sets inhabited by numbers. Formalized Mathematics, 11(4): 341-347, 2003. [35] Michał J. Trybulec. Integers. Formalized

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Torsion Part of ℤ-module

structures. Formalized Mathematics , 9( 3 ):559–564, 2001. [28] Christoph Schwarzweller. The ring of integers, Euclidean rings and modulo integers. Formalized Mathematics , 8( 1 ):29–34, 1999. [29] Andrzej Trybulec. Domains and their Cartesian products. Formalized Mathematics , 1( 1 ): 115–122, 1990. [30] Michał J. Trybulec. Integers. Formalized Mathematics , 1( 3 ):501–505, 1990. [31] Wojciech A. Trybulec. Operations on subspaces in real linear space. Formalized Mathematics , 1( 2 ):395–399, 1990. [32] Wojciech A. Trybulec. Vectors in

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Isomorphisms from the Space of Multilinear Operators

] Hiroyuki Okazaki, Noboru Endou, and Yasunari Shidama. Cartesian products of family of real linear spaces. Formalized Mathematics , 19( 1 ):51–59, 2011. doi:10.2478/v10037-011-0009-2. [9] Laurent Schwartz. Théorie des ensembles et topologie, tome 1. Analyse . Hermann, 1997. [10] Laurent Schwartz. Calcul différentiel, tome 2. Analyse . Hermann, 1997. [11] Kosaku Yoshida. Functional Analysis . Springer, 1980.

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Invertible Operators on Banach Spaces

] Kazuhisa Nakasho, Yuichi Futa, and Yasunari Shidama. Implicit function theorem. Part I. Formalized Mathematics , 25( 4 ):269–281, 2017. doi:10.1515/forma-2017-0026. [5] Hiroyuki Okazaki, Noboru Endou, and Yasunari Shidama. Cartesian products of family of real linear spaces. Formalized Mathematics , 19( 1 ):51–59, 2011. doi:10.2478/v10037-011-0009-2. [6] Laurent Schwartz. Théorie des ensembles et topologie, tome 1. Analyse . Hermann, 1997. [7] Laurent Schwartz. Calcul différentiel, tome 2. Analyse . Hermann, 1997. [8] Yasunari Shidama. Banach

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