Theorem 3.1: P5 holds in the real projective plane.
Def: A configuration is a set of points and lines such that two distinct points lie on at most one line.
Example: Desargues configuration.
Theorem 3.6: P5 holds in any projective 3-space, where we do not assume that all points necessarily lie in a plane. In particular, P5 holds for any plane that lies in a projective 3- space.
Rmk: There are competing definitions for projective 3-space. In some of them, Thm 3.6 is false (Hartvigson).
\pi* is called the dual projective plane of \pi.
Converse of Desargues' axiom.
Principle of Duality
Proposition 3.11: If a projective plane contains a line with n+1 points, then all lines have n+1 points and there are n2 + n + 1 points in the plane.
Proposition 3.12: If a projective plane contains a point with a pencil of n+1 lines, then all points have pencils of n+1 lines and there are n2 + n + 1 lines in the plane.
Remark: The dual of the dual of a plane is the original plane. The dual of a plane need not be isomorphic to the original plane, but this is true for the real projective plane.