70 research outputs found
Some Properties of p-Groups and Commutative p-Groups
This article describes some properties of p-groups and some properties of commutative p-groups.Liang Xiquan - Qingdao University of Science and Technology, ChinaLi Dailu - Qingdao University of Science and Technology, ChinaGrzegorz 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.RafaĆ Kwiatek. Factorial and Newton coefficients. Formalized Mathematics, 1(5):887-890, 1990.Marco Riccardi. The Sylow theorems. Formalized Mathematics, 15(3):159-165, 2007, doi:10.2478/v10037-007-0018-3.Dariusz Surowik. Cyclic groups and some of their properties - part I. Formalized Mathematics, 2(5):623-627, 1991.Wojciech A. Trybulec. Classes of conjugation. Normal subgroups. Formalized Mathematics, 1(5):955-962, 1990.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.Wojciech A. Trybulec. Commutator and center of a group. Formalized Mathematics, 2(4):461-466, 1991.Wojciech A. Trybulec. Lattice of subgroups of a group. Frattini subgroup. Formalized Mathematics, 2(1):41-47, 1991.Wojciech A. Trybulec and MichaĆ J. Trybulec. Homomorphisms and isomorphisms of groups. Quotient group. Formalized Mathematics, 2(4):573-578, 1991.Zinaida Trybulec. Properties of subsets. Formalized Mathematics, 1(1):67-71, 1990
On Rough Subgroup of a Group
This article describes a rough subgroup with respect to a normal
subgroup of a group, and some properties of the lower and the upper approximations
in a group.Liang Xiquan - Qingdao University of Science and Technology, ChinaLi Dailu - Qingdao University of Science and Technology, ChinaWojciech A. Trybulec. Classes of conjugation. Normal subgroups. Formalized Mathematics, 1(5):955-962, 1990.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.Wojciech A. Trybulec. Lattice of subgroups of a group. Frattini subgroup. Formalized Mathematics, 2(1):41-47, 1991.Zinaida Trybulec. Properties of subsets. Formalized Mathematics, 1(1):67-71, 1990
Formalization of Integral Linear Space
In this article, we formalize integral linear spaces, that is a linear space with integer coefficients. Integral linear spaces are necessary for lattice problems, LLL (Lenstra-Lenstra-LovĂĄsz) base reduction algorithm that outputs short lattice base and cryptographic systems with lattice [8].Futa Yuichi - Shinshu University, Nagano, JapanOkazaki Hiroyuki - Shinshu University, Nagano, JapanShidama Yasunari - Shinshu University, Nagano, JapanGrzegorz 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.CzesĆaw ByliĆski. Partial functions. Formalized Mathematics, 1(2):357-367, 1990.Noboru Endou, Takashi Mitsuishi, and Yasunari Shidama. Dimension of real unitary space. Formalized Mathematics, 11(1):23-28, 2003.Krzysztof Hryniewiecki. Basic properties of real numbers. Formalized Mathematics, 1(1):35-40, 1990.JarosĆaw Kotowicz. Real sequences and basic operations on them. Formalized Mathematics, 1(2):269-272, 1990.Daniele Micciancio and Shafi Goldwasser. Complexity of lattice problems: A cryptographic perspective (the international series in engineering and computer science). 2002.Andrzej Trybulec. Binary operations applied to functions. Formalized Mathematics, 1(2):329-334, 1990.Andrzej Trybulec. Domains and their Cartesian products. Formalized Mathematics, 1(1):115-122, 1990.MichaĆ J. Trybulec. Integers. Formalized Mathematics, 1(3):501-505, 1990.Wojciech A. Trybulec. Basis of real linear space. Formalized Mathematics, 1(5):847-850, 1990.Wojciech A. Trybulec. Linear combinations in real linear space. Formalized Mathematics, 1(3):581-588, 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 defined on sets. Formalized Mathematics, 1(1):181-186, 1990.Hiroshi Yamazaki and Yasunari Shidama. Algebra of vector functions. Formalized Mathematics, 3(2):171-175, 1992
The Geometric Interior in Real Linear Spaces
We introduce the notions of the geometric interior and the centre of mass for subsets of real linear spaces. We prove a number of theorems
concerning these notions which are used in the theory of abstract simplicial complexes.Institute of Informatics, University of BiaĆystok, PolandGrzegorz Bancerek. Cardinal numbers. Formalized Mathematics, 1(2):377-382, 1990.Grzegorz Bancerek and Krzysztof Hryniewiecki. Segments of natural numbers and finite sequences. Formalized Mathematics, 1(1):107-114, 1990.CzesĆaw ByliĆski. Functions and their basic properties. Formalized Mathematics, 1(1):55-65, 1990.CzesĆaw ByliĆski. Functions from a set to a set. Formalized Mathematics, 1(1):153-164, 1990.Agata DarmochwaĆ. Finite sets. Formalized Mathematics, 1(1):165-167, 1990.Noboru Endou, Takashi Mitsuishi, and Yasunari Shidama. Convex sets and convex combinations. Formalized Mathematics, 11(1):53-58, 2003.Krzysztof Hryniewiecki. Basic properties of real numbers. Formalized Mathematics, 1(1):35-40, 1990.JarosĆaw Kotowicz. Real sequences and basic operations on them. Formalized Mathematics, 1(2):269-272, 1990.Beata Padlewska. Families of sets. Formalized Mathematics, 1(1):147-152, 1990.Karol PÄ
k. Affine independence in vector spaces. Formalized Mathematics, 18(1):87-93, 2010, doi: 10.2478/v10037-010-0012-z.Andrzej Trybulec. Domains and their Cartesian products. Formalized Mathematics, 1(1):115-122, 1990.Wojciech A. Trybulec. Linear combinations in real linear space. Formalized Mathematics, 1(3):581-588, 1990.Wojciech A. Trybulec. Partially ordered sets. Formalized Mathematics, 1(2):313-319, 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 defined on sets. Formalized Mathematics, 1(1):181-186, 1990
The Real Vector Spaces of Finite Sequences are Finite Dimensional
In this paper we show the finite dimensionality of real linear spaces with their carriers equal Rn. We also give the standard basis of such spaces. For the set Rn we introduce the concepts of linear manifold subsets and orthogonal subsets. The cardinality of orthonormal basis of discussed spaces is proved to equal n.Yatsuka Nakamura - Shinshu University Nagano, JapanNagato Oya - Shinshu University Nagano, JapanYasunari Shidama - Shinshu University Nagano, JapanArtur KorniĆowicz - Institute of Computer Science, University of BiaĆystok, Sosnowa 64, 15-887 BiaĆystok, Polan
Valuation Theory. Part I
In the article we introduce a valuation function over a field [1]. Ring of non negative elements and its ideal of positive elements have been also defined.Bancerek Grzegorz - BiaĆystok Technical University, PolandKobayashi Hidetsune - Department of Mathematics College of Science and Technology, Nihon University, 8 Kanda Surugadai Chiyoda-ku, 101-8308 Tokyo, JapanKorniĆowicz Artur - Institute of Informatics, University of BiaĆystok, Sosnowa 64, 15-887 BiaĆystok, PolandEmil Artin. Algebraic Numbers and Algebraic Functions. Gordon and Breach Science Publishers, 1994.Jonathan Backer, Piotr Rudnicki, and Christoph Schwarzweller. Ring ideals. Formalized Mathematics, 9(3):565-582, 2001.Grzegorz 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.JĂłzef BiaĆas. Properties of fields. Formalized Mathematics, 1(5):807-812, 1990.CzesĆaw ByliĆski. Binary operations. Formalized Mathematics, 1(1):175-180, 1990.CzesĆaw ByliĆski. Functions and their basic properties. Formalized Mathematics, 1(1):55-65, 1990.CzesĆaw ByliĆski. Functions from a set to a set. Formalized Mathematics, 1(1):153-164, 1990.CzesĆaw ByliĆski. Partial functions. Formalized Mathematics, 1(2):357-367, 1990.CzesĆaw ByliĆski. Some basic properties of sets. Formalized Mathematics, 1(1):47-53, 1990.Artur KorniĆowicz. Quotient rings. Formalized Mathematics, 13(4):573-576, 2005.Eugeniusz Kusak, Wojciech LeoĆczuk, and MichaĆ Muzalewski. Abelian groups, fields and vector spaces. Formalized Mathematics, 1(2):335-342, 1990.MichaĆ Muzalewski. Construction of rings and left-, right-, and bi-modules over a ring. Formalized Mathematics, 2(1):3-11, 1991.Andrzej Trybulec. Domains and their Cartesian products. Formalized Mathematics, 1(1):115-122, 1990.MichaĆ J. Trybulec. Integers. Formalized Mathematics, 1(3):501-505, 1990.Wojciech A. Trybulec. Groups. Formalized Mathematics, 1(5):821-827, 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
Set of Points on Elliptic Curve in Projective Coordinates
In this article, we formalize a set of points on an elliptic curve over GF(p). Elliptic curve cryptography [10], whose security is based on a difficulty of discrete logarithm problem of elliptic curves, is important for information security.Futa Yuichi - Shinshu University, Nagano, JapanOkazaki Hiroyuki - Shinshu University, Nagano, JapanShidama Yasunari - Shinshu University, Nagano, JapanGrzegorz 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.JĂłzef BiaĆas. Group and field definitions. Formalized Mathematics, 1(3):433-439, 1990.CzesĆaw ByliĆski. Functions and their basic properties. Formalized Mathematics, 1(1):55-65, 1990.CzesĆaw ByliĆski. Functions from a set to a set. Formalized Mathematics, 1(1):153-164, 1990.CzesĆaw ByliĆski. Some basic properties of sets. Formalized Mathematics, 1(1):47-53, 1990.Agata DarmochwaĆ. Finite sets. Formalized Mathematics, 1(1):165-167, 1990.Krzysztof Hryniewiecki. Basic properties of real numbers. Formalized Mathematics, 1(1):35-40, 1990.G. Seroussi I. Blake and N. Smart. Elliptic Curves in Cryptography. Number 265 in London Mathematical Society Lecture Note Series. Cambridge University Press, 1999.Eugeniusz Kusak, Wojciech LeoĆczuk, and MichaĆ Muzalewski. Abelian groups, fields and vector spaces. Formalized Mathematics, 1(2):335-342, 1990.RafaĆ Kwiatek. Factorial and Newton coefficients. Formalized Mathematics, 1(5):887-890, 1990.Konrad Raczkowski and PaweĆ Sadowski. Equivalence relations and classes of abstraction. Formalized Mathematics, 1(3):441-444, 1990.Christoph Schwarzweller. The ring of integers, euclidean rings and modulo integers. Formalized Mathematics, 8(1):29-34, 1999.Christoph Schwarzweller. The binomial theorem for algebraic structures. Formalized Mathematics, 9(3):559-564, 2001.Andrzej Trybulec. Domains and their Cartesian products. Formalized Mathematics, 1(1):115-122, 1990.Andrzej Trybulec. Tuples, projections and Cartesian products. Formalized Mathematics, 1(1):97-105, 1990.MichaĆ J. Trybulec. Integers. Formalized Mathematics, 1(3):501-505, 1990.Wojciech A. Trybulec. Groups. Formalized Mathematics, 1(5):821-827, 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 and Anna Zalewska. Properties of binary relations. Formalized Mathematics, 1(1):85-89, 1990
Cartesian Products of Family of Real Linear Spaces
In this article we introduced the isomorphism mapping between cartesian products of family of linear spaces [4]. Those products had been formalized by two different ways, i.e., the way using the functor [:X, Y:] and ones using the functor "product". By the same way, the isomorphism mapping was defined between Cartesian products of family of linear normed spaces also.Okazaki Hiroyuki - Shinshu University, Nagano, JapanEndou Noboru - Nagano National College of Technology, JapanShidama Yasunari - Shinshu University, Nagano, JapanGrzegorz 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.Nicolas Bourbaki. Topological vector spaces: Chapters 1-5. Springer, 1981.CzesĆaw ByliĆski. Binary operations. Formalized Mathematics, 1(1):175-180, 1990.CzesĆaw ByliĆski. Finite sequences and tuples of elements of a non-empty sets. Formalized Mathematics, 1(3):529-536, 1990.CzesĆaw ByliĆski. Functions and their basic properties. Formalized Mathematics, 1(1):55-65, 1990.CzesĆaw ByliĆski. Functions from a set to a set. Formalized Mathematics, 1(1):153-164, 1990.CzesĆaw ByliĆski. Some basic properties of sets. Formalized Mathematics, 1(1):47-53, 1990.CzesĆaw ByliĆski. The sum and product of finite sequences of real numbers. Formalized Mathematics, 1(4):661-668, 1990.Agata DarmochwaĆ. The Euclidean space. Formalized Mathematics, 2(4):599-603, 1991.Noboru Endou, Yasunari Shidama, and Keiichi Miyajima. The product space of real normed spaces and its properties. Formalized Mathematics, 15(3):81-85, 2007, doi:10.2478/v10037-007-0010-y.Beata Padlewska and Agata DarmochwaĆ. Topological spaces and continuous functions. Formalized Mathematics, 1(1):223-230, 1990.Jan PopioĆek. Real normed space. Formalized Mathematics, 2(1):111-115, 1991.Yasunari Shidama. Banach space of bounded linear operators. Formalized Mathematics, 12(1):39-48, 2004.Andrzej Trybulec. Domains and their Cartesian products. Formalized Mathematics, 1(1):115-122, 1990.Wojciech A. Trybulec. Pigeon hole principle. Formalized Mathematics, 1(3):575-579, 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
Planes and Spheres as Topological Manifolds. Stereographic Projection
The goal of this article is to show some examples of topological manifolds: planes and spheres in Euclidean space. In doing it, the article introduces the stereographic projection [25].Via del Pero 102, 54038 Montignoso, ItalyGrzegorz 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. König's theorem. Formalized Mathematics, 1(3):589-593, 1990.Grzegorz Bancerek. The ordinal numbers. Formalized Mathematics, 1(1):91-96, 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.CzesĆaw ByliĆski. Binary operations. Formalized Mathematics, 1(1):175-180, 1990.CzesĆaw ByliĆski. Finite sequences and tuples of elements of a non-empty sets. Formalized Mathematics, 1(3):529-536, 1990.CzesĆaw ByliĆski. Functions and their basic properties. Formalized Mathematics, 1(1):55-65, 1990.CzesĆaw ByliĆski. Functions from a set to a set. Formalized Mathematics, 1(1):153-164, 1990.CzesĆaw ByliĆski. Partial functions. Formalized Mathematics, 1(2):357-367, 1990.CzesĆaw ByliĆski. Some basic properties of sets. Formalized Mathematics, 1(1):47-53, 1990.CzesĆaw ByliĆski. The sum and product of finite sequences of real numbers. Formalized Mathematics, 1(4):661-668, 1990.Agata DarmochwaĆ. Families of subsets, subspaces and mappings in topological spaces. Formalized Mathematics, 1(2):257-261, 1990.Agata DarmochwaĆ. Finite sets. Formalized Mathematics, 1(1):165-167, 1990.Agata DarmochwaĆ. The Euclidean space. Formalized Mathematics, 2(4):599-603, 1991.Agata DarmochwaĆ and Yatsuka Nakamura. Metric spaces as topological spaces - fundamental concepts. Formalized Mathematics, 2(4):605-608, 1991.Krzysztof Hryniewiecki. Basic properties of real numbers. Formalized Mathematics, 1(1):35-40, 1990.Katarzyna Jankowska. Matrices. Abelian group of matrices. Formalized Mathematics, 2(4):475-480, 1991.StanisĆawa Kanas, Adam Lecko, and Mariusz Startek. Metric spaces. Formalized Mathematics, 1(3):607-610, 1990.Artur KorniĆowicz and Yasunari Shidama. Intersections of intervals and balls in En/T Formalized Mathematics, 12(3):301-306, 2004.Artur KorniĆowicz and Yasunari Shidama. Some properties of circles on the plane. Formalized Mathematics, 13(1):117-124, 2005.JarosĆaw Kotowicz. Real sequences and basic operations on them. Formalized Mathematics, 1(2):269-272, 1990.Eugeniusz Kusak, Wojciech LeoĆczuk, and MichaĆ Muzalewski. Abelian groups, fields and vector spaces. Formalized Mathematics, 1(2):335-342, 1990.John M. Lee. Introduction to Topological Manifolds. Springer-Verlag, New York Berlin Heidelberg, 2000.Robert Milewski. Bases of continuous lattices. Formalized Mathematics, 7(2):285-294, 1998.Yatsuka Nakamura, Artur KorniĆowicz, Nagato Oya, and Yasunari Shidama. The real vector spaces of finite sequences are finite dimensional. Formalized Mathematics, 17(1):1-9, 2009, doi:10.2478/v10037-009-0001-2.Henryk Oryszczyszyn and Krzysztof PraĆŒmowski. Real functions spaces. Formalized Mathematics, 1(3):555-561, 1990.Beata Padlewska. Families of sets. Formalized Mathematics, 1(1):147-152, 1990.Beata Padlewska. Locally connected spaces. Formalized Mathematics, 2(1):93-96, 1991.Beata Padlewska and Agata DarmochwaĆ. Topological spaces and continuous functions. Formalized Mathematics, 1(1):223-230, 1990.Karol PÄ
k. Basic properties of metrizable topological spaces. Formalized Mathematics, 17(3):201-205, 2009, doi: 10.2478/v10037-009-0024-8.Marco Riccardi. The definition of topological manifolds. Formalized Mathematics, 19(1):41-44, 2011, doi: 10.2478/v10037-011-0007-4.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. Basis of real linear space. Formalized Mathematics, 1(5):847-850, 1990.Wojciech A. Trybulec. Linear combinations in real linear space. Formalized Mathematics, 1(3):581-588, 1990.Wojciech A. Trybulec. Subspaces and cosets of subspaces in real linear space. Formalized Mathematics, 1(2):297-301, 1990.Wojciech A. Trybulec. Vectors in real linear space. Formalized Mathematics, 1(2):291-296, 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.Mariusz ƻynel and Adam Guzowski. T0 topological spaces. Formalized Mathematics, 5(1):75-77, 1996
Operations of Points on Elliptic Curve in Projective Coordinates
In this article, we formalize operations of points on an elliptic curve over GF(p). Elliptic curve cryptography [7], whose security is based on a difficulty of discrete logarithm problem of elliptic curves, is important for information security. We prove that the two operations of points: compellProjCo and addellProjCo are unary and binary operations of a point over the elliptic curve.Futa Yuichi - Shinshu University, Nagano, JapanOkazaki Hiroyuki - Shinshu University, Nagano, JapanMizushima Daichi - Shinshu University, Nagano, JapanShidama Yasunari - Shinshu University, Nagano, JapanGrzegorz Bancerek. The ordinal numbers. Formalized Mathematics, 1(1):91-96, 1990.CzesĆaw ByliĆski. Binary operations. Formalized Mathematics, 1(1):175-180, 1990.CzesĆaw ByliĆski. Functions and their basic properties. Formalized Mathematics, 1(1):55-65, 1990.CzesĆaw ByliĆski. Functions from a set to a set. Formalized Mathematics, 1(1):153-164, 1990.CzesĆaw ByliĆski. Some basic properties of sets. Formalized Mathematics, 1(1):47-53, 1990.Yuichi Futa, Hiroyuki Okazaki, and Yasunari Shidama. Set of points on elliptic curve in projective coordinates. Formalized Mathematics, 19(3):131-138, 2011, doi: 10.2478/v10037-011-0021-6.G. Seroussi I. Blake and N. Smart. Elliptic Curves in Cryptography. Number 265 in London Mathematical Society Lecture Note Series. Cambridge University Press, 1999.Eugeniusz Kusak, Wojciech LeoĆczuk, and MichaĆ Muzalewski. Abelian groups, fields and vector spaces. Formalized Mathematics, 1(2):335-342, 1990.RafaĆ Kwiatek. Factorial and Newton coefficients. Formalized Mathematics, 1(5):887-890, 1990.RafaĆ Kwiatek and Grzegorz Zwara. The divisibility of integers and integer relative primes. Formalized Mathematics, 1(5):829-832, 1990.Christoph Schwarzweller. The binomial theorem for algebraic structures. Formalized Mathematics, 9(3):559-564, 2001.Andrzej Trybulec. Domains and their Cartesian products. Formalized Mathematics, 1(1):115-122, 1990.Andrzej Trybulec. Tuples, projections and Cartesian products. Formalized Mathematics, 1(1):97-105, 1990.MichaĆ J. Trybulec. Integers. Formalized Mathematics, 1(3):501-505, 1990.Wojciech A. Trybulec. Groups. Formalized Mathematics, 1(5):821-827, 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
- âŠ