Journal of Formalized Mathematics
Volume 11, 1999
University of Bialystok
Copyright (c) 1999 Association of Mizar Users

Properties of Left and Right Components


Artur Kornilowicz
University of Bialystok
This paper was written while the author visited Shinshu University, winter 1999.

MML Identifier: GOBRD14

The terminology and notation used in this paper have been introduced in the following articles [28] [9] [34] [35] [7] [8] [6] [2] [16] [31] [1] [17] [24] [10] [19] [18] [5] [4] [3] [30] [27] [26] [25] [13] [14] [15] [20] [22] [29] [12] [21] [33] [32] [23] [11]

Contents (PDF format)

  1. Components
  2. Go-Boards
  3. LeftComp and RightComp

Acknowledgments

I would like to thank Professor Yatsuka Nakamura for his help in the preparation of the article.

Bibliography

[1] Grzegorz Bancerek. The fundamental properties of natural numbers. Journal of Formalized Mathematics, 1, 1989.
[2] Grzegorz Bancerek. The ordinal numbers. Journal of Formalized Mathematics, 1, 1989.
[3] Grzegorz Bancerek. Countable sets and Hessenberg's theorem. Journal of Formalized Mathematics, 2, 1990.
[4] Grzegorz Bancerek. K\"onig's theorem. Journal of Formalized Mathematics, 2, 1990.
[5] Grzegorz Bancerek and Krzysztof Hryniewiecki. Segments of natural numbers and finite sequences. Journal of Formalized Mathematics, 1, 1989.
[6] Czeslaw Bylinski. Binary operations. Journal of Formalized Mathematics, 1, 1989.
[7] Czeslaw Bylinski. Functions and their basic properties. Journal of Formalized Mathematics, 1, 1989.
[8] Czeslaw Bylinski. Functions from a set to a set. Journal of Formalized Mathematics, 1, 1989.
[9] Czeslaw Bylinski. Some basic properties of sets. Journal of Formalized Mathematics, 1, 1989.
[10] Czeslaw Bylinski. The modification of a function by a function and the iteration of the composition of a function. Journal of Formalized Mathematics, 2, 1990.
[11] Czeslaw Bylinski. Gauges. Journal of Formalized Mathematics, 11, 1999.
[12] Czeslaw Bylinski and Piotr Rudnicki. Bounding boxes for compact sets in $\calE^2$. Journal of Formalized Mathematics, 9, 1997.
[13] Agata Darmochwal. Compact spaces. Journal of Formalized Mathematics, 1, 1989.
[14] Agata Darmochwal. The Euclidean space. Journal of Formalized Mathematics, 3, 1991.
[15] Agata Darmochwal and Yatsuka Nakamura. The topological space $\calE^2_\rmT$. Arcs, line segments and special polygonal arcs. Journal of Formalized Mathematics, 3, 1991.
[16] Krzysztof Hryniewiecki. Basic properties of real numbers. Journal of Formalized Mathematics, 1, 1989.
[17] Katarzyna Jankowska. Matrices. Abelian group of matrices. Journal of Formalized Mathematics, 3, 1991.
[18] Stanislawa Kanas, Adam Lecko, and Mariusz Startek. Metric spaces. Journal of Formalized Mathematics, 2, 1990.
[19] Jaroslaw Kotowicz. Convergent real sequences. Upper and lower bound of sets of real numbers. Journal of Formalized Mathematics, 1, 1989.
[20] Jaroslaw Kotowicz and Yatsuka Nakamura. Introduction to Go-Board --- part I. Journal of Formalized Mathematics, 4, 1992.
[21] Yatsuka Nakamura and Czeslaw Bylinski. Extremal properties of vertices on special polygons, part I. Journal of Formalized Mathematics, 6, 1994.
[22] Yatsuka Nakamura and Andrzej Trybulec. Decomposing a Go-Board into cells. Journal of Formalized Mathematics, 7, 1995.
[23] Yatsuka Nakamura, Andrzej Trybulec, and Czeslaw Bylinski. Bounded domains and unbounded domains. Journal of Formalized Mathematics, 11, 1999.
[24] Takaya Nishiyama and Yasuho Mizuhara. Binary arithmetics. Journal of Formalized Mathematics, 5, 1993.
[25] Beata Padlewska. Connected spaces. Journal of Formalized Mathematics, 1, 1989.
[26] Beata Padlewska and Agata Darmochwal. Topological spaces and continuous functions. Journal of Formalized Mathematics, 1, 1989.
[27] Konrad Raczkowski and Pawel Sadowski. Topological properties of subsets in real numbers. Journal of Formalized Mathematics, 2, 1990.
[28] Andrzej Trybulec. Tarski Grothendieck set theory. Journal of Formalized Mathematics, Axiomatics, 1989.
[29] Andrzej Trybulec. Left and right component of the complement of a special closed curve. Journal of Formalized Mathematics, 7, 1995.
[30] Andrzej Trybulec. On the decomposition of finite sequences. Journal of Formalized Mathematics, 7, 1995.
[31] Andrzej Trybulec and Czeslaw Bylinski. Some properties of real numbers operations: min, max, square, and square root. Journal of Formalized Mathematics, 1, 1989.
[32] Andrzej Trybulec and Yatsuka Nakamura. On the order on a special polygon. Journal of Formalized Mathematics, 9, 1997.
[33] Andrzej Trybulec and Yatsuka Nakamura. On the rectangular finite sequences of the points of the plane. Journal of Formalized Mathematics, 9, 1997.
[34] Zinaida Trybulec. Properties of subsets. Journal of Formalized Mathematics, 1, 1989.
[35] Edmund Woronowicz. Relations and their basic properties. Journal of Formalized Mathematics, 1, 1989.

Received May 5, 1999

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