Volume 9, 1997

University of Bialystok

Copyright (c) 1997 Association of Mizar Users

**Yatsuka Nakamura**- Shinshu University, Nagano
**Andrzej Trybulec**- University of Bialystok

- Various types of the intermediate value theorem ( [14]) are proved. For their special cases, the Bolzano theorem is also proved. Using such a theorem, it is shown that if a curve is a simple closed curve, then it is not horizontally degenerated, neither is it vertically degenerated.

- Intermediate Value Theorems and Bolzano Theorem
- Simple Closed Curves Are Not Flat

- [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 and Piotr Rudnicki.
Bounding boxes for compact sets in $\calE^2$.
*Journal of Formalized Mathematics*, 9, 1997. - [6]
Agata Darmochwal.
Compact spaces.
*Journal of Formalized Mathematics*, 1, 1989. - [7]
Agata Darmochwal.
The Euclidean space.
*Journal of Formalized Mathematics*, 3, 1991. - [8]
Agata Darmochwal and Yatsuka Nakamura.
Metric spaces as topological spaces --- fundamental concepts.
*Journal of Formalized Mathematics*, 3, 1991. - [9]
Agata Darmochwal and Yatsuka Nakamura.
The topological space $\calE^2_\rmT$. Simple closed curves.
*Journal of Formalized Mathematics*, 3, 1991. - [10]
Stanislawa Kanas, Adam Lecko, and Mariusz Startek.
Metric spaces.
*Journal of Formalized Mathematics*, 2, 1990. - [11]
Beata Padlewska.
Connected spaces.
*Journal of Formalized Mathematics*, 1, 1989. - [12]
Beata Padlewska and Agata Darmochwal.
Topological spaces and continuous functions.
*Journal of Formalized Mathematics*, 1, 1989. - [13]
Konrad Raczkowski and Pawel Sadowski.
Topological properties of subsets in real numbers.
*Journal of Formalized Mathematics*, 2, 1990. - [14] Georgi E. Shilov, editor. \em Elementary Real and Complex Analysis(English translation, translated by Richard A. Silverman). The MIT Press, 1973.
- [15]
Andrzej Trybulec.
Tarski Grothendieck set theory.
*Journal of Formalized Mathematics*, Axiomatics, 1989. - [16]
Andrzej Trybulec.
A Borsuk theorem on homotopy types.
*Journal of Formalized Mathematics*, 3, 1991. - [17]
Andrzej Trybulec.
Subsets of real numbers.
*Journal of Formalized Mathematics*, Addenda, 2003. - [18]
Zinaida Trybulec.
Properties of subsets.
*Journal of Formalized Mathematics*, 1, 1989. - [19]
Edmund Woronowicz.
Relations and their basic properties.
*Journal of Formalized Mathematics*, 1, 1989.

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