Consider two distinct lines [i]l [/i]and [i]m [/i]in a projective geometry. Now, consider some point P not on either line. For every point X on [i]l[/i], the line PX intersects line [i]m [/i]at some point Y[i]. [/i]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 [i]m, [/i]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 [i]m[/i] and every point on [i]l[/i].This new line XPY creates the one-to-one correspondence between points on line [i]l[/i] and points on line [i]m. [/i]

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. [br]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.