the proof of the parallel theorem

The Parallel Theorem
If L is a line in affine geometry and P is a point not on L,
then there is a unique line through P not meeting L.


We can regard z =1 as the affine plane. Each point P or line L on this arises
from a p-point P* or p-line L*, respectively.

Suppose that M* is a p-line intersecting z=1. Then M* meets L* in a unique
p-point R*. Observe that the p-line L* meets z=0 in a unique p-point T*.
Thus, the p-lines M* with R* not on z=1 are precisely those through T*.
Given the p-point P*, there is a unique p-line M* through P* and T*.

Reverting to the affine plane, M is the unique line through P which does not
meet L.

postlude page