Journal of Formalized Mathematics
Volume 2, 1990
University of Bialystok
Copyright (c) 1990
Association of Mizar Users
The Collinearity Structure

Wojciech Skaba

Nicolaus Copernicus University, Torun

Supported by RPBP.III24.B5.
Summary.

The text includes basic axioms and theorems concerning
the collinearity structure based on Wanda Szmielew [2],
pp. 1820. Collinearity
is defined as a relation on Cartesian product
$\mizleftcart S, S, S \mizrightcart$ of set $S$.
The basic text is preceeded with a few auxiliary theorems
(e.g: ternary relation). Then come the two basic axioms of the collinearity
structure: A1.1.1 and A1.1.2 and a few theorems. Another axiom:
Aks dim, which states that there exist at least 3 noncollinear points,
excludes the trivial structures (i.e. pairs
$\llangle S, \mizleftcart S, S, S \mizrightcart\rrangle$).
Following it the notion of a line is included and several additional
theorems are appended.
MML Identifier:
COLLSP
The terminology and notation used in this paper have been
introduced in the following articles
[4]
[1]
[5]
[3]
Contents (PDF format)
Bibliography
 [1]
Czeslaw Bylinski.
Some basic properties of sets.
Journal of Formalized Mathematics,
1, 1989.
 [2]
Wanda Szmielew.
\em From Affine to Euclidean Geometry, volume 27.
PWN  D.Reidel Publ. Co., Warszawa  Dordrecht, 1983.
 [3]
Andrzej Trybulec.
Domains and their Cartesian products.
Journal of Formalized Mathematics,
1, 1989.
 [4]
Andrzej Trybulec.
Tarski Grothendieck set theory.
Journal of Formalized Mathematics,
Axiomatics, 1989.
 [5]
Zinaida Trybulec.
Properties of subsets.
Journal of Formalized Mathematics,
1, 1989.
Received May 9, 1990
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