Volume 15, 2003

University of Bialystok

Copyright (c) 2003 Association of Mizar Users

**Akihiro Kubo**- Shinshu University, Nagano
**Yatsuka Nakamura**- Shinshu University, Nagano

- Two transformations between the complex space and 2-dimensional Euclidian topological space are defined. By them, the concept of argument is induced to 2-dimensional vectors using argument of complex number. Similarly, the concept of an angle is introduced using the angle of two complex numbers. The concept of a triangle and related concepts are also defined in $n$-dimensional Euclidian topological spaces.

Contents (PDF format)

- [1]
Kanchun and Yatsuka Nakamura.
The inner product of finite sequences and of points of $n$-dimensional topological space.
*Journal of Formalized Mathematics*, 15, 2003. - [2]
Grzegorz Bancerek and Krzysztof Hryniewiecki.
Segments of natural numbers and finite sequences.
*Journal of Formalized Mathematics*, 1, 1989. - [3]
Czeslaw Bylinski.
Functions and their basic properties.
*Journal of Formalized Mathematics*, 1, 1989. - [4]
Czeslaw Bylinski.
Functions from a set to a set.
*Journal of Formalized Mathematics*, 1, 1989. - [5]
Czeslaw Bylinski.
The complex numbers.
*Journal of Formalized Mathematics*, 2, 1990. - [6]
Czeslaw Bylinski.
The sum and product of finite sequences of real numbers.
*Journal of Formalized Mathematics*, 2, 1990. - [7]
Wenpai Chang and Yatsuka Nakamura.
Inner products and angles of complex numbers.
*Journal of Formalized Mathematics*, 15, 2003. - [8]
Library Committee.
Introduction to arithmetic.
*Journal of Formalized Mathematics*, Addenda, 2003. - [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]
Krzysztof Hryniewiecki.
Basic properties of real numbers.
*Journal of Formalized Mathematics*, 1, 1989. - [13]
Jaroslaw Kotowicz.
Real sequences and basic operations on them.
*Journal of Formalized Mathematics*, 1, 1989. - [14]
Beata Padlewska and Agata Darmochwal.
Topological spaces and continuous functions.
*Journal of Formalized Mathematics*, 1, 1989. - [15]
Konrad Raczkowski and Pawel Sadowski.
Topological properties of subsets in real numbers.
*Journal of Formalized Mathematics*, 2, 1990. - [16]
Agnieszka Sakowicz, Jaroslaw Gryko, and Adam Grabowski.
Sequences in $\calE^N_\rmT$.
*Journal of Formalized Mathematics*, 6, 1994. - [17]
Andrzej Trybulec.
Tarski Grothendieck set theory.
*Journal of Formalized Mathematics*, Axiomatics, 1989. - [18]
Andrzej Trybulec.
Subsets of real numbers.
*Journal of Formalized Mathematics*, Addenda, 2003. - [19]
Andrzej Trybulec and Czeslaw Bylinski.
Some properties of real numbers operations: min, max, square, and square root.
*Journal of Formalized Mathematics*, 1, 1989. - [20]
Zinaida Trybulec.
Properties of subsets.
*Journal of Formalized Mathematics*, 1, 1989. - [21]
Edmund Woronowicz.
Relations and their basic properties.
*Journal of Formalized Mathematics*, 1, 1989. - [22]
Yuguang Yang and Yasunari Shidama.
Trigonometric functions and existence of circle ratio.
*Journal of Formalized Mathematics*, 10, 1998.

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