@MISC{ORTSP_1.MIZ,
  AUTHOR = {Kusak, Eugeniusz and Leo\'nczuk, Wojciech and Muzalewski, Micha{\l}},
  TITLE = {Construction of a bilinear symmetric form
in orthogonal vector space},
  NOTE = {Supported by RPBP.III-24.C6.},
  DAY = {23},
  MONTH = {November},
  YEAR = {1989},
  ADDRESS1 = {Warsaw University\\Bia{\l}ystok},
  ADDRESS2 = {Warsaw University\\Bia{\l}ystok},
  ADDRESS3 = {Warsaw University\\Bia{\l}ystok},
  SUMMARY = {  In this text we present unpublished results by Eu\-ge\-niusz Ku\-sak and
Wojciech Leo\'nczuk. They contain an axiomatic description of the class of all
spaces $\langle V$; $\perp_\xi \rangle$, where $V$ is a vector
space over a field F, $\xi: V \times V \to F$ is a bilinear symmetric form
  i.e. $\xi(x,y) = \xi(y,x)$
   and $x \perp_\xi y$ iff $\xi(x,y) = 0$
for $x$, $y \in V$.
   They also contain an effective construction of bilinear symmetric
form $\xi$ for given orthogonal space $\langle V$; $\perp \rangle$
such that $\perp = \perp_\xi$. The
basic tool used in this method is the notion of orthogonal projection
J$(a,b,x)$ for $a,b,x \in V$.
  We should stress the fact that axioms of orthogonal and symplectic
spaces differ only by one axiom, namely:
 $x\perp y+\varepsilon z \>\&\> y\perp z+\varepsilon x \Rightarrow
z\perp x+\varepsilon y.$
For $\varepsilon=-1$ we get the axiom on three perpendiculars characterizing
orthogonal geometry. For $\varepsilon=+1$ we get the axiom characterizing
symplectic geometry - see \cite{SYMSP_1.ABS}.}}

