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

## Analytical Metric Affine Spaces and Planes

Henryk Oryszczyszyn
Warsaw University, Bialystok
Krzysztof Prazmowski
Warsaw University, Bialystok

### Summary.

We introduce relations of orthogonality of vectors and of orthogonality of segments (considered as pairs of vectors) in real linear space of dimension two. This enables us to show an example of (in fact anisotropic and satisfying theorem on three perpendiculars) metric affine space (and plane as well). These two types of objects are defined formally as "Mizar" modes. They are to be understood as structures consisting of a point universe and two binary relations on segments - a parallelity relation and orthogonality relation, satisfying appropriate axioms. With every such structure we correlate a structure obtained as a reduct of the given one to the parallelity relation only. Some relationships between metric affine spaces and their affine parts are proved; they enable us to use "affine" facts and constructions in investigating metric affine geometry. We define the notions of line, parallelity of lines and two derived relations of orthogonality: between segments and lines, and between lines. Some basic properties of the introduced notions are proved.

Supported by RPBP.III-24.C2.

#### MML Identifier: ANALMETR

The terminology and notation used in this paper have been introduced in the following articles [8] [2] [10] [7] [3] [5] [11] [9] [6] [4] [1]

Contents (PDF format)

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