Journal of Formalized Mathematics
Volume 15, 2003
University of Bialystok
Copyright (c) 2003 Association of Mizar Users

General Fashoda Meet Theorem for Unit Circle and Square


Yatsuka Nakamura
Shinshu University, Nagano

Summary.

Here we will prove Fashoda meet theorem for the unit circle and for a square, when 4 points on the boundary are ordered cyclically. Also, the concepts of general rectangle and general circle are defined.

MML Identifier: JGRAPH_6

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

Contents (PDF format)

  1. Preliminaries
  2. General Fashoda Theorem for Unit Circle
  3. General Rectangles and Circles
  4. Order of Points on Rectangle
  5. General Fashoda Theorem for Square

Bibliography

[1] Grzegorz Bancerek. The ordinal numbers. Journal of Formalized Mathematics, 1, 1989.
[2] Grzegorz Bancerek and Krzysztof Hryniewiecki. Segments of natural numbers and finite sequences. Journal of Formalized Mathematics, 1, 1989.
[3] Leszek Borys. Paracompact and metrizable spaces. Journal of Formalized Mathematics, 3, 1991.
[4] Czeslaw Bylinski. Functions and their basic properties. Journal of Formalized Mathematics, 1, 1989.
[5] Czeslaw Bylinski. Functions from a set to a set. Journal of Formalized Mathematics, 1, 1989.
[6] Czeslaw Bylinski and Piotr Rudnicki. Bounding boxes for compact sets in $\calE^2$. Journal of Formalized Mathematics, 9, 1997.
[7] Agata Darmochwal. Compact spaces. Journal of Formalized Mathematics, 1, 1989.
[8] Agata Darmochwal. Families of subsets, subspaces and mappings in topological spaces. Journal of Formalized Mathematics, 1, 1989.
[9] Agata Darmochwal. The Euclidean space. Journal of Formalized Mathematics, 3, 1991.
[10] Agata Darmochwal and Yatsuka Nakamura. Metric spaces as topological spaces --- fundamental concepts. Journal of Formalized Mathematics, 3, 1991.
[11] Agata Darmochwal and Yatsuka Nakamura. The topological space $\calE^2_\rmT$. Arcs, line segments and special polygonal arcs. Journal of Formalized Mathematics, 3, 1991.
[12] Agata Darmochwal and Yatsuka Nakamura. The topological space $\calE^2_\rmT$. Simple closed curves. Journal of Formalized Mathematics, 3, 1991.
[13] Krzysztof Hryniewiecki. Basic properties of real numbers. Journal of Formalized Mathematics, 1, 1989.
[14] Stanislawa Kanas, Adam Lecko, and Mariusz Startek. Metric spaces. Journal of Formalized Mathematics, 2, 1990.
[15] Artur Kornilowicz. The ordering of points on a curve, part III. Journal of Formalized Mathematics, 14, 2002.
[16] Yatsuka Nakamura. On Outside Fashoda Meet Theorem. Journal of Formalized Mathematics, 13, 2001.
[17] Yatsuka Nakamura. On the simple closed curve property of the circle and the Fashoda Meet Theorem for it. Journal of Formalized Mathematics, 13, 2001.
[18] Yatsuka Nakamura and Andrzej Trybulec. A decomposition of simple closed curves and the order of their points. Journal of Formalized Mathematics, 9, 1997.
[19] Beata Padlewska and Agata Darmochwal. Topological spaces and continuous functions. Journal of Formalized Mathematics, 1, 1989.
[20] Konrad Raczkowski and Pawel Sadowski. Topological properties of subsets in real numbers. Journal of Formalized Mathematics, 2, 1990.
[21] Agnieszka Sakowicz, Jaroslaw Gryko, and Adam Grabowski. Sequences in $\calE^N_\rmT$. Journal of Formalized Mathematics, 6, 1994.
[22] Andrzej Trybulec. Tarski Grothendieck set theory. Journal of Formalized Mathematics, Axiomatics, 1989.
[23] Andrzej Trybulec. Subsets of real numbers. Journal of Formalized Mathematics, Addenda, 2003.
[24] Andrzej Trybulec and Czeslaw Bylinski. Some properties of real numbers operations: min, max, square, and square root. Journal of Formalized Mathematics, 1, 1989.
[25] Wojciech A. Trybulec. Pigeon hole principle. Journal of Formalized Mathematics, 2, 1990.
[26] Zinaida Trybulec. Properties of subsets. Journal of Formalized Mathematics, 1, 1989.
[27] Edmund Woronowicz. Relations and their basic properties. Journal of Formalized Mathematics, 1, 1989.

Received February 25, 2003


[ Download a postscript version, MML identifier index, Mizar home page]