Volume 13, 2001

University of Bialystok

Copyright (c) 2001 Association of Mizar Users

**Andrzej Trybulec**- University of Bialystok

- The main goal was to prove two facts: \begin{itemize} \itemsep-3pt \item the gauge is the Go-Board of a corresponding cage, \item the left components of the complement of the curve determined by a cage are monotonic wrt the index of the approximation. \end{itemize} Some auxiliary facts are proved, too. At the end the new notion needed for the internal approximation are defined and some useful lemmas are proved.

This work has been partially supported by CALCULEMUS grant HPRN-CT-2000-00102.

- Preliminaries
- Transforming Finite Sets to Finite Sequences
- On the Construction of Go-Boards
- More about Go-Boards
- More about Gauges
- More about Cages
- Preparing the Internal Approximation

- [1]
Grzegorz Bancerek.
Cardinal numbers.
*Journal of Formalized Mathematics*, 1, 1989. - [2]
Grzegorz Bancerek.
The fundamental properties of natural numbers.
*Journal of Formalized Mathematics*, 1, 1989. - [3]
Grzegorz Bancerek.
The ordinal numbers.
*Journal of Formalized Mathematics*, 1, 1989. - [4]
Grzegorz Bancerek.
Countable sets and Hessenberg'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]
Leszek Borys.
Paracompact and metrizable spaces.
*Journal of Formalized Mathematics*, 3, 1991. - [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.
Partial functions.
*Journal of Formalized Mathematics*, 1, 1989. - [10]
Czeslaw Bylinski.
Some basic properties of sets.
*Journal of Formalized Mathematics*, 1, 1989. - [11]
Czeslaw Bylinski.
Finite sequences and tuples of elements of a non-empty sets.
*Journal of Formalized Mathematics*, 2, 1990. - [12]
Czeslaw Bylinski.
The sum and product of finite sequences of real numbers.
*Journal of Formalized Mathematics*, 2, 1990. - [13]
Czeslaw Bylinski.
Some properties of restrictions of finite sequences.
*Journal of Formalized Mathematics*, 7, 1995. - [14]
Czeslaw Bylinski.
Gauges.
*Journal of Formalized Mathematics*, 11, 1999. - [15]
Czeslaw Bylinski.
Some properties of cells on go board.
*Journal of Formalized Mathematics*, 11, 1999. - [16]
Czeslaw Bylinski and Piotr Rudnicki.
Bounding boxes for compact sets in $\calE^2$.
*Journal of Formalized Mathematics*, 9, 1997. - [17]
Czeslaw Bylinski and Mariusz Zynel.
Cages - the external approximation of Jordan's curve.
*Journal of Formalized Mathematics*, 11, 1999. - [18]
Agata Darmochwal.
Compact spaces.
*Journal of Formalized Mathematics*, 1, 1989. - [19]
Agata Darmochwal.
Finite sets.
*Journal of Formalized Mathematics*, 1, 1989. - [20]
Agata Darmochwal.
The Euclidean space.
*Journal of Formalized Mathematics*, 3, 1991. - [21]
Agata Darmochwal and Yatsuka Nakamura.
The topological space $\calE^2_\rmT$. Arcs, line segments and special polygonal arcs.
*Journal of Formalized Mathematics*, 3, 1991. - [22]
Krzysztof Hryniewiecki.
Basic properties of real numbers.
*Journal of Formalized Mathematics*, 1, 1989. - [23]
Katarzyna Jankowska.
Matrices. Abelian group of matrices.
*Journal of Formalized Mathematics*, 3, 1991. - [24]
Stanislawa Kanas, Adam Lecko, and Mariusz Startek.
Metric spaces.
*Journal of Formalized Mathematics*, 2, 1990. - [25]
Artur Kornilowicz, Robert Milewski, Adam Naumowicz, and Andrzej Trybulec.
Gauges and cages. Part I.
*Journal of Formalized Mathematics*, 12, 2000. - [26]
Jaroslaw Kotowicz and Yatsuka Nakamura.
Introduction to Go-Board --- part I.
*Journal of Formalized Mathematics*, 4, 1992. - [27]
Jaroslaw Kotowicz and Yatsuka Nakamura.
Introduction to Go-Board --- part II.
*Journal of Formalized Mathematics*, 4, 1992. - [28]
Beata Madras.
On the concept of the triangulation.
*Journal of Formalized Mathematics*, 7, 1995. - [29]
Yatsuka Nakamura and Czeslaw Bylinski.
Extremal properties of vertices on special polygons, part I.
*Journal of Formalized Mathematics*, 6, 1994. - [30]
Yatsuka Nakamura and Jaroslaw Kotowicz.
The Jordan's property for certain subsets of the plane.
*Journal of Formalized Mathematics*, 4, 1992. - [31]
Yatsuka Nakamura and Roman Matuszewski.
Reconstructions of special sequences.
*Journal of Formalized Mathematics*, 8, 1996. - [32]
Yatsuka Nakamura and Andrzej Trybulec.
Decomposing a Go-Board into cells.
*Journal of Formalized Mathematics*, 7, 1995. - [33]
Yatsuka Nakamura, Andrzej Trybulec, and Czeslaw Bylinski.
Bounded domains and unbounded domains.
*Journal of Formalized Mathematics*, 11, 1999. - [34]
Takaya Nishiyama and Yasuho Mizuhara.
Binary arithmetics.
*Journal of Formalized Mathematics*, 5, 1993. - [35]
Beata Padlewska.
Connected spaces.
*Journal of Formalized Mathematics*, 1, 1989. - [36]
Beata Padlewska and Agata Darmochwal.
Topological spaces and continuous functions.
*Journal of Formalized Mathematics*, 1, 1989. - [37]
Andrzej Trybulec.
Tarski Grothendieck set theory.
*Journal of Formalized Mathematics*, Axiomatics, 1989. - [38]
Andrzej Trybulec.
Left and right component of the complement of a special closed curve.
*Journal of Formalized Mathematics*, 7, 1995. - [39]
Andrzej Trybulec.
On the decomposition of finite sequences.
*Journal of Formalized Mathematics*, 7, 1995. - [40]
Andrzej Trybulec.
Subsets of real numbers.
*Journal of Formalized Mathematics*, Addenda, 2003. - [41]
Andrzej Trybulec and Yatsuka Nakamura.
On the order on a special polygon.
*Journal of Formalized Mathematics*, 9, 1997. - [42]
Michal J. Trybulec.
Integers.
*Journal of Formalized Mathematics*, 2, 1990. - [43]
Wojciech A. Trybulec.
Partially ordered sets.
*Journal of Formalized Mathematics*, 1, 1989. - [44]
Wojciech A. Trybulec.
Pigeon hole principle.
*Journal of Formalized Mathematics*, 2, 1990. - [45]
Wojciech A. Trybulec and Grzegorz Bancerek.
Kuratowski - Zorn lemma.
*Journal of Formalized Mathematics*, 1, 1989. - [46]
Zinaida Trybulec.
Properties of subsets.
*Journal of Formalized Mathematics*, 1, 1989. - [47]
Edmund Woronowicz.
Relations and their basic properties.
*Journal of Formalized Mathematics*, 1, 1989. - [48]
Edmund Woronowicz.
Relations defined on sets.
*Journal of Formalized Mathematics*, 1, 1989. - [49]
Edmund Woronowicz and Anna Zalewska.
Properties of binary relations.
*Journal of Formalized Mathematics*, 1, 1989. - [50]
Miroslaw Wysocki and Agata Darmochwal.
Subsets of topological spaces.
*Journal of Formalized Mathematics*, 1, 1989.

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