Differentiability of Polynomials over Reals

In this article, we formalize in the Mizar system [3] the notion of the derivative of polynomials over the field of real numbers [4]. To define it, we use the derivative of functions between reals and reals [9].Institute of Informatics, University of, Białystok, PolandGrzegorz Bancerek. The fundamental properties of natural numbers. Formalized Mathematics, 1(1):41-46, 1990.Grzegorz Bancerek and Andrzej Trybulec. Miscellaneous facts about functions. Formalized Mathematics, 5(4):485-492, 1996.Adam Grabowski, Artur Korniłowicz, and Adam Naumowicz. Four decades of Mizar. Journal of Automated Reasoning, 55(3):191-198, 2015. doi: 10.1007/s10817-015-9345-1.Kazimierz Kuratowski. Rachunek rózniczkowy i całkowy - funkcje jednej zmiennej. Biblioteka Matematyczna. PWN - Warszawa (in polish), 1964.Robert Milewski. The evaluation of polynomials. Formalized Mathematics, 9(2):391-395, 2001.Robert Milewski. Fundamental theorem of algebra. Formalized Mathematics, 9(3):461-470, 2001.Michał Muzalewski and Lesław W. Szczerba. Construction of finite sequences over ring and left-, right-, and bi-modules over a ring. Formalized Mathematics, 2(1):97-104, 1991.Konrad Raczkowski. Integer and rational exponents. Formalized Mathematics, 2(1):125-130, 1991.Konrad Raczkowski and Paweł Sadowski. Real function differentiability. Formalized Mathematics, 1(4):797-801, 1990.Andrzej Trybulec. Binary operations applied to functions. Formalized Mathematics, 1 (2):329-334, 1990.Michał J. Trybulec. Integers. Formalized Mathematics, 1(3):501-505, 1990

The Correspondence Between n-dimensional Euclidean Space and the Product of n Real Lines

In the article we prove that a family of open n-hypercubes is a basis of n-dimensional Euclidean space. The equality of the space and the product of n real lines has been proven.Institute of Informatics, University of Białystok, Sosnowa 64, 15-887 Białystok, PolandGrzegorz Bancerek. The fundamental properties of natural numbers. Formalized Mathematics, 1(1):41-46, 1990.Grzegorz Bancerek. König's theorem. Formalized Mathematics, 1(3):589-593, 1990.Grzegorz Bancerek. The ordinal numbers. Formalized Mathematics, 1(1):91-96, 1990.Grzegorz Bancerek and Krzysztof Hryniewiecki. Segments of natural numbers and finite sequences. Formalized Mathematics, 1(1):107-114, 1990.Leszek Borys. Paracompact and metrizable spaces. Formalized Mathematics, 2(4):481-485, 1991.Czesław Byliński. The complex numbers. Formalized Mathematics, 1(3):507-513, 1990.Czesław Byliński. Finite sequences and tuples of elements of a non-empty sets. Formalized Mathematics, 1(3):529-536, 1990.Czesław Byliński. Functions and their basic properties. Formalized Mathematics, 1(1):55-65, 1990.Czesław Byliński. The sum and product of finite sequences of real numbers. Formalized Mathematics, 1(4):661-668, 1990.Agata Darmochwał. The Euclidean space. Formalized Mathematics, 2(4):599-603, 1991.Agata Darmochwał and Yatsuka Nakamura. Metric spaces as topological spaces - fundamental concepts. Formalized Mathematics, 2(4):605-608, 1991.Jarosław Gryko. Injective spaces. Formalized Mathematics, 7(1):57-62, 1998.Stanisława Kanas, Adam Lecko, and Mariusz Startek. Metric spaces. Formalized Mathematics, 1(3):607-610, 1990.Artur Korniłowicz. Arithmetic operations on functions from sets into functional sets. Formalized Mathematics, 17(1):43-60, 2009, doi:10.2478/v10037-009-0005-y.Beata Madras. Product of family of universal algebras. Formalized Mathematics, 4(1):103-108, 1993.Beata Padlewska and Agata Darmochwał. Topological spaces and continuous functions. Formalized Mathematics, 1(1):223-230, 1990.Konrad Raczkowski and Paweł Sadowski. Topological properties of subsets in real numbers. Formalized Mathematics, 1(4):777-780, 1990.Andrzej Trybulec. Binary operations applied to functions. Formalized Mathematics, 1(2):329-334, 1990.Andrzej Trybulec and Czesław Byliński. Some properties of real numbers. Formalized Mathematics, 1(3):445-449, 1990.Zinaida Trybulec. Properties of subsets. Formalized Mathematics, 1(1):67-71, 1990.Edmund Woronowicz. Relations and their basic properties. Formalized Mathematics, 1(1):73-83, 1990.Edmund Woronowicz. Relations defined on sets. Formalized Mathematics, 1(1):181-186, 1990

Cayley's Theorem

The article formalizes the Cayley's theorem saying that every group G is isomorphic to a subgroup of the symmetric group on G.Institute of Informatics, University of Białystok, Sosnowa 64, 15-887 Białystok, PolandGrzegorz Bancerek. The fundamental properties of natural numbers. Formalized Mathematics, 1(1):41-46, 1990.Grzegorz Bancerek. Monoids. Formalized Mathematics, 3(2):213-225, 1992.Grzegorz Bancerek and Krzysztof Hryniewiecki. Segments of natural numbers and finite sequences. Formalized Mathematics, 1(1):107-114, 1990.Czesław Byliński. Functions and their basic properties. Formalized Mathematics, 1(1):55-65, 1990.Czesław Byliński. Functions from a set to a set. Formalized Mathematics, 1(1):153-164, 1990.Agata Darmochwał. Finite sets. Formalized Mathematics, 1(1):165-167, 1990.Katarzyna Jankowska. Transpose matrices and groups of permutations. Formalized Mathematics, 2(5):711-717, 1991.Artur Korniłowicz. The definition and basic properties of topological groups. Formalized Mathematics, 7(2):217-225, 1998.Andrzej Trybulec. Classes of independent partitions. Formalized Mathematics, 9(3):623-625, 2001.Wojciech A. Trybulec. Groups. Formalized Mathematics, 1(5):821-827, 1990.Wojciech A. Trybulec and Michał J. Trybulec. Homomorphisms and isomorphisms of groups. Quotient group. Formalized Mathematics, 2(4):573-578, 1991

The Borsuk-Ulam Theorem

The Borsuk-Ulam theorem about antipodals is proven, [18, pp. 32-33].This work has been supported by the Polish Ministry of Science and Higher Education project “Managing a Large Repository of Computer-verified Mathematical Knowledge” (N N519 385136)Korniłowicz Artur - Institute of Informatics, University of Białystok, Sosnowa 64, 15-887 Białystok, PolandRiccardi Marco - Via del Pero 102, 54038 Montignoso, ItalyGrzegorz Bancerek. The fundamental properties of natural numbers. Formalized Mathematics, 1(1):41-46, 1990.Grzegorz Bancerek. König’s theorem. Formalized Mathematics, 1(3):589-593, 1990.Grzegorz Bancerek. The ordinal numbers. Formalized Mathematics, 1(1):91-96, 1990.Grzegorz Bancerek and Krzysztof Hryniewiecki. Segments of natural numbers and finite sequences. Formalized Mathematics, 1(1):107-114, 1990.Czesław Bylinski. The complex numbers. Formalized Mathematics, 1(3):507-513, 1990.Czesław Bylinski. Functions and their basic properties. Formalized Mathematics, 1(1):55-65, 1990.Czesław Bylinski. Functions from a set to a set. Formalized Mathematics, 1(1):153-164, 1990.Czesław Bylinski. The modification of a function by a function and the iteration of the composition of a function. Formalized Mathematics, 1(3):521-527, 1990.Czesław Bylinski. Partial functions. Formalized Mathematics, 1(2):357-367, 1990.Czesław Bylinski. Some basic properties of sets. Formalized Mathematics, 1(1):47-53, 1990.Czesław Bylinski. The sum and product of finite sequences of real numbers. Formalized Mathematics, 1(4):661-668, 1990.Agata Darmochwał. Families of subsets, subspaces and mappings in topological spaces. Formalized Mathematics, 1(2):257-261, 1990.Agata Darmochwał. The Euclidean space. Formalized Mathematics, 2(4):599-603, 1991.Agata Darmochwał and Yatsuka Nakamura. Metric spaces as topological spaces - fundamental concepts. Formalized Mathematics, 2(4):605-608, 1991.Adam Grabowski. Introduction to the homotopy theory. Formalized Mathematics, 6(4):449-454, 1997.Adam Grabowski. On the subcontinua of a real line. Formalized Mathematics, 11(3):313-322, 2003.Jarosław Gryko. Injective spaces. Formalized Mathematics, 7(1):57-62, 1998.Allen Hatcher. Algebraic Topology. Cambridge University Press, 2002.Krzysztof Hryniewiecki. Basic properties of real numbers. Formalized Mathematics, 1(1):35-40, 1990.Kanchun, Hiroshi Yamazaki, and Yatsuka Nakamura. Cross products and tripple vector products in 3-dimensional Euclidean space. Formalized Mathematics, 11(4):381-383, 2003.Artur Korniłowicz. Arithmetic operations on functions from sets into functional sets. Formalized Mathematics, 17(1):43-60, 2009, doi:10.2478/v10037-009-0005-y.Artur Korniłowicz. On the continuity of some functions. Formalized Mathematics, 18(3):175-183, 2010, doi: 10.2478/v10037-010-0020-z.Artur Korniłowicz and Yasunari Shidama. Intersections of intervals and balls in EnT. Formalized Mathematics, 12(3):301-306, 2004.Artur Korniłowicz and Yasunari Shidama. Some properties of circles on the plane. Formalized Mathematics, 13(1):117-124, 2005.Artur Korniłowicz, Yasunari Shidama, and Adam Grabowski. The fundamental group. Formalized Mathematics, 12(3):261-268, 2004.Akihiro Kubo and Yatsuka Nakamura. Angle and triangle in Euclidian topological space. Formalized Mathematics, 11(3):281-287, 2003.Adam Naumowicz and Grzegorz Bancerek. Homeomorphisms of Jordan curves. Formalized Mathematics, 13(4):477-480, 2005.Beata Padlewska. Connected spaces. Formalized Mathematics, 1(1):239-244, 1990.Beata Padlewska and Agata Darmochwał. Topological spaces and continuous functions. Formalized Mathematics, 1(1):223-230, 1990.Konrad Raczkowski and Paweł Sadowski. Real function continuity. Formalized Mathematics, 1(4):787-791, 1990.Konrad Raczkowski and Paweł Sadowski. Topological properties of subsets in real numbers. Formalized Mathematics, 1(4):777-780, 1990.Marco Riccardi and Artur Korniłowicz. Fundamental group of n-sphere for n ≥ 2. Formalized Mathematics, 20(2):97-104, 2012, doi: 10.2478/v10037-012-0013-1.Piotr Rudnicki and Andrzej Trybulec. Abian’s fixed point theorem. Formalized Mathematics, 6(3):335-338, 1997.Andrzej Trybulec. Binary operations applied to functions. Formalized Mathematics, 1(2):329-334, 1990.Andrzej Trybulec. Enumerated sets. Formalized Mathematics, 1(1):25-34, 1990.Andrzej Trybulec. A Borsuk theorem on homotopy types. Formalized Mathematics, 2(4):535-545, 1991.Andrzej Trybulec and Czesław Bylinski. Some properties of real numbers. Formalized Mathematics, 1(3):445-449, 1990.Michał J. Trybulec. Integers. Formalized Mathematics, 1(3):501-505, 1990.Wojciech A. Trybulec. Vectors in real linear space. Formalized Mathematics, 1(2):291-296, 1990.Zinaida Trybulec. Properties of subsets. Formalized Mathematics, 1(1):67-71, 1990.Edmund Woronowicz. Relations and their basic properties. Formalized Mathematics, 1(1):73-83, 1990.Edmund Woronowicz. Relations defined on sets. Formalized Mathematics, 1(1):181-186, 1990.Yuguang Yang and Yasunari Shidama. Trigonometric functions and existence of circle ratio. Formalized Mathematics, 7(2):255-263, 1998

Fundamental Group of n-sphere for n ≥ 2

Triviality of fundamental groups of spheres of dimension greater than 1 is proven, [17].This work has been supported by the Polish Ministry of Science and Higher Education project “Managing a Large Repository of Computer-verified Mathematical Knowledge” (N N519 385136).Riccardi Marco - Via del Pero 102, 54038 Montignoso, ItalyKorniłowicz Artur - Institute of Informatics, University of Białystok, Sosnowa 64, 15-887 Białystok, PolandGrzegorz Bancerek. The fundamental properties of natural numbers. Formalized Mathematics, 1(1):41-46, 1990.Grzegorz Bancerek. The ordinal numbers. Formalized Mathematics, 1(1):91-96, 1990.Grzegorz Bancerek and Krzysztof Hryniewiecki. Segments of natural numbers and finite sequences. Formalized Mathematics, 1(1):107-114, 1990.Czesław Bylinski. Binary operations. Formalized Mathematics, 1(1):175-180, 1990.Czesław Bylinski. Functions and their basic properties. Formalized Mathematics, 1(1):55-65, 1990.Czesław Bylinski. Functions from a set to a set. Formalized Mathematics, 1(1):153-164, 1990.Czesław Bylinski. Partial functions. Formalized Mathematics, 1(2):357-367, 1990.Czesław Bylinski. Some basic properties of sets. Formalized Mathematics, 1(1):47-53, 1990.Agata Darmochwał. The Euclidean space. Formalized Mathematics, 2(4):599-603, 1991.Agata Darmochwał and Yatsuka Nakamura. Metric spaces as topological spaces - fundamental concepts. Formalized Mathematics, 2(4):605-608, 1991.Adam Grabowski. Introduction to the homotopy theory. Formalized Mathematics, 6(4):449-454, 1997.Adam Grabowski and Artur Korniłowicz. Algebraic properties of homotopies. Formalized Mathematics, 12(3):251-260, 2004.Artur Korniłowicz. The fundamental group of convex subspaces of EnT. Formalized Mathematics, 12(3):295-299, 2004.Artur Korniłowicz. On the isomorphism of fundamental groups. Formalized Mathematics, 12(3):391-396, 2004.Artur Korniłowicz and Yasunari Shidama. Intersections of intervals and balls in EnT. Formalized Mathematics, 12(3):301-306, 2004.Artur Korniłowicz, Yasunari Shidama, and Adam Grabowski. The fundamental group. Formalized Mathematics, 12(3):261-268, 2004.John M. Lee. Introduction to Topological Manifolds. Springer-Verlag, New York Berlin Heidelberg, 2000.Beata Padlewska and Agata Darmochwał. Topological spaces and continuous functions. Formalized Mathematics, 1(1):223-230, 1990.Konrad Raczkowski and Paweł Sadowski. Equivalence relations and classes of abstraction. Formalized Mathematics, 1(3):441-444, 1990.Konrad Raczkowski and Paweł Sadowski. Topological properties of subsets in real numbers. Formalized Mathematics, 1(4):777-780, 1990.Marco Riccardi. The definition of topological manifolds. Formalized Mathematics, 19(1):41-44, 2011, doi: 10.2478/v10037-011-0007-4.Marco Riccardi. Planes and spheres as topological manifolds. Stereographic projection. Formalized Mathematics, 20(1):41-45, 2012, doi: 10.2478/v10037-012-0006-0.Zinaida Trybulec. Properties of subsets. Formalized Mathematics, 1(1):67-71, 1990.Edmund Woronowicz. Relations and their basic properties. Formalized Mathematics, 1(1):73-83, 1990.Edmund Woronowicz. Relations defined on sets. Formalized Mathematics, 1(1):181-186, 1990

Introduction to Liouville Numbers

The article defines Liouville numbers, originally introduced by Joseph Liouville in 1844 [17] as an example of an object which can be approximated “quite closely” by a sequence of rational numbers. A real number x is a Liouville number iff for every positive integer n, there exist integers p and q such that q > 1 and 0 < x − p q < 1 q n . It is easy to show that all Liouville numbers are irrational. Liouville constant, which is also defined formally, is the first transcendental (not algebraic) number. It is defined in Section 6 quite generally as the sum X∞ k=1 ak b k! for a finite sequence {ak}k∈N and b ∈ N. Based on this definition, we also introduced the so-called Liouville number as L = X∞ k=1 10−k! = 0.110001000000000000000001 . . . , substituting in the definition of L(ak, b) the constant sequence of 1’s and b = 10. Another important examples of transcendental numbers are e and π [7], [13], [6]. At the end, we show that the construction of an arbitrary Lioville constant satisfies the properties of a Liouville number [12], [1]. We show additionally, that the set of all Liouville numbers is infinite, opening the next item from Abad and Abad’s list of “Top 100 Theorems”. We show also some preliminary constructions linking real sequences and finite sequences, where summing formulas are involved. In the Mizar [14] proof, we follow closely https: //en.wikipedia.org/wiki/Liouville_number. The aim is to show that all Liouville numbers are transcendental.Grabowski Adam - Institute of Informatics, University of Białystok, Białystok, PolandKorniłowicz Artur - Institute of Informatics, University of Białystok, Białystok, PolandTom M. Apostol. Modular Functions and Dirichlet Series in Number Theory. Springer- Verlag, 2nd edition, 1997.Grzegorz Bancerek. Cardinal numbers. Formalized Mathematics, 1(2):377-382, 1990.Grzegorz Bancerek. The fundamental properties of natural numbers. Formalized Mathematics, 1(1):41-46, 1990.Grzegorz Bancerek and Krzysztof Hryniewiecki. Segments of natural numbers and finite sequences. Formalized Mathematics, 1(1):107-114, 1990.Grzegorz Bancerek and Piotr Rudnicki. Two programs for SCM. Part I - preliminaries. Formalized Mathematics, 4(1):69-72, 1993.Sophie Bernard, Yves Bertot, Laurence Rideau, and Pierre-Yves Strub. Formal proofs of transcendence for e and π as an application of multivariate and symmetric polynomials. In Jeremy Avigad and Adam Chlipala, editors, Proceedings of the 5th ACM SIGPLAN Conference on Certified Programs and Proofs, pages 76-87. ACM, 2016.Jesse Bingham. Formalizing a proof that e is transcendental. Journal of Formalized Reasoning, 4:71-84, 2011.Czesław Byliński. Functions and their basic properties. Formalized Mathematics, 1(1): 55-65, 1990.Czesław Byliński. Functions from a set to a set. Formalized Mathematics, 1(1):153-164, 1990.Czesław Byliński. The modification of a function by a function and the iteration of the composition of a function. Formalized Mathematics, 1(3):521-527, 1990.Czesław Byliński. Some basic properties of sets. Formalized Mathematics, 1(1):47-53, 1990.J.H. Conway and R.K. Guy. The Book of Numbers. Springer-Verlag, 1996.Manuel Eberl. Liouville numbers. Archive of Formal Proofs, December 2015. http://isa-afp.org/entries/Liouville_Numbers.shtml, Formal proof development.Adam Grabowski, Artur Korniłowicz, and Adam Naumowicz. Four decades of Mizar. Journal of Automated Reasoning, 55(3):191-198, 2015.Jarosław Kotowicz. Real sequences and basic operations on them. Formalized Mathematics, 1(2):269-272, 1990.Rafał Kwiatek. Factorial and Newton coefficients. Formalized Mathematics, 1(5):887-890, 1990.Joseph Liouville. Nouvelle démonstration d’un théorème sur les irrationnelles algébriques, inséré dans le Compte Rendu de la dernière séance. Compte Rendu Acad. Sci. Paris, Sér.A (18):910–911, 1844.Jan Popiołek. Some properties of functions modul and signum. Formalized Mathematics, 1(2):263-264, 1990.Konrad Raczkowski. Integer and rational exponents. Formalized Mathematics, 2(1):125-130, 1991.Konrad Raczkowski and Andrzej Nedzusiak. Real exponents and logarithms. Formalized Mathematics, 2(2):213-216, 1991.Michał J. Trybulec. Integers. Formalized Mathematics, 1(3):501-505, 1990.Wojciech A. Trybulec. Non-contiguous substrings and one-to-one finite sequences. Formalized Mathematics, 1(3):569-573, 1990.Wojciech A. Trybulec. Binary operations on finite sequences. Formalized Mathematics, 1 (5):979-981, 1990.Edmund Woronowicz. Relations and their basic properties. Formalized Mathematics, 1 (1):73-83, 1990

Commutativeness of Fundamental Groups of Topological Groups

In this article we prove that fundamental groups based at the unit point of topological groups are commutative [11].Institute of Informatics University of Białystok Sosnowa 64, 15-887 Białystok PolandGrzegorz Bancerek. Monoids. Formalized Mathematics, 3(2):213-225, 1992.Józef Białas. Group and field definitions. Formalized Mathematics, 1(3):433-439, 1990.Czesław Bylinski. Binary operations. Formalized Mathematics, 1(1):175-180, 1990.Czesław Bylinski. Functions and their basic properties. Formalized Mathematics, 1(1): 55-65, 1990.Czesław Bylinski. Functions from a set to a set. Formalized Mathematics, 1(1):153-164, 1990.Czesław Bylinski. Partial functions. Formalized Mathematics, 1(2):357-367, 1990.Czesław Bylinski. Some basic properties of sets. Formalized Mathematics, 1(1):47-53, 1990.Agata Darmochwał and Yatsuka Nakamura. Metric spaces as topological spaces - fundamental concepts. Formalized Mathematics, 2(4):605-608, 1991.Adam Grabowski. Introduction to the homotopy theory. Formalized Mathematics, 6(4): 449-454, 1997.Adam Grabowski and Artur Korniłowicz. Algebraic properties of homotopies. Formalized Mathematics, 12(3):251-260, 2004.Allen Hatcher. Algebraic Topology. Cambridge University Press, 2002.Artur Korniłowicz. The fundamental group of convex subspaces of EnT. Formalized Mathematics, 12(3):295-299, 2004.Artur Korniłowicz. The definition and basic properties of topological groups. Formalized Mathematics, 7(2):217-225, 1998.Artur Korniłowicz and Yasunari Shidama. Some properties of circles on the plane. Formalized Mathematics, 13(1):117-124, 2005.Artur Korniłowicz, Yasunari Shidama, and Adam Grabowski. The fundamental group. Formalized Mathematics, 12(3):261-268, 2004.Beata Padlewska. Locally connected spaces. Formalized Mathematics, 2(1):93-96, 1991.Beata Padlewska and Agata Darmochwał. Topological spaces and continuous functions. Formalized Mathematics, 1(1):223-230, 1990.Konrad Raczkowski and Paweł Sadowski. Topological properties of subsets in real numbers. Formalized Mathematics, 1(4):777-780, 1990.Andrzej Trybulec. A Borsuk theorem on homotopy types. Formalized Mathematics, 2(4): 535-545, 1991.Andrzej Trybulec. Binary operations applied to functions. Formalized Mathematics, 1 (2):329-334, 1990.Andrzej Trybulec. On the sets inhabited by numbers. Formalized Mathematics, 11(4): 341-347, 2003.Wojciech A. Trybulec. Groups. Formalized Mathematics, 1(5):821-827, 1990.Wojciech A. Trybulec. Subgroup and cosets of subgroups. Formalized Mathematics, 1(5): 855-864, 1990.Zinaida Trybulec. Properties of subsets. Formalized Mathematics, 1(1):67-71, 1990.Edmund Woronowicz. Relations and their basic properties. Formalized Mathematics, 1 (1):73-83, 1990.Edmund Woronowicz. Relations defined on sets. Formalized Mathematics, 1(1):181-186, 1990

Products in Categories without Uniqueness of cod and dom

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].This work has been supported by the Polish Ministry of Science and Higher Education project “Managing a Large Repository of Computer-verified Mathematical Knowledge” (N N519 385136).Institute of Informatics, University of Białystok, Sosnowa 64, 15-887 Białystok PolandGrzegorz Bancerek. König’s theorem. Formalized Mathematics, 1(3):589-593, 1990.Czesław Bylinski. Functions and their basic properties. Formalized Mathematics, 1(1):55-65, 1990.Czesław Bylinski. Functions from a set to a set. Formalized Mathematics, 1(1):153-164, 1990.Czesław Bylinski. Partial functions. Formalized Mathematics, 1(2):357-367, 1990.Czesław Bylinski. Some basic properties of sets. Formalized Mathematics, 1(1):47-53, 1990.Beata Madras. Basic properties of objects and morphisms. Formalized Mathematics, 6(3):329-334, 1997.Zbigniew Semadeni and Antoni Wiweger. Wstęp do teorii kategorii i funktorów, volume 45 of Biblioteka Matematyczna. PWN, Warszawa, 1978.Andrzej Trybulec. Binary operations applied to functions. Formalized Mathematics, 1(2):329-334, 1990.Andrzej Trybulec. Many sorted sets. Formalized Mathematics, 4(1):15-22, 1993.Andrzej Trybulec. Categories without uniqueness of cod and dom. Formalized Mathematics, 5(2):259-267, 1996.Zinaida Trybulec. Properties of subsets. Formalized Mathematics, 1(1):67-71, 1990.Edmund Woronowicz. Relations and their basic properties. Formalized Mathematics, 1(1):73-83, 1990