Proof Lemma 7.1
In a projective plane, the points on one line can be put into one-to-one correspondence with the points on any other line.
Consider two distinct lines l and m in a projective geometry. Now, consider some point P not on either line. For every point X on l, the line PX intersects line m at some point Y. We know this from Theorem 7.7 which states that any two distinct lines have exactly one point in common. Thus, the line through points P and X shares a point with line m, creating the line XPY. PA2 tells us that any two distinct points has exactly one line in common. Thus, the line XPY should contain every point on m and every point on l.This new line XPY creates the one-to-one correspondence between points on line l and points on line m.
Dual of Lemma 7.1: The lines on one point can be put into one-to-one correspondence with the lines on any other point.
Consider two distinct points P and Q. Dual Axiom 3 tells us that these two points have at least one line in common. If the two points have more than one line in common, then Dual Axiom 2 is contradicted. Thus, the points P and Q must have exactly one in common, call it l. The line l thus has a one-to-one correspondence with itself.
We also know from Dual Axiom 2 that any line, m, through P and not through Q must share a point, call it R, with line, n, through Q and not through P. The point R, thus creates a one-to-one correspondence between two distinct lines that are not on both P and Q.