Proof: Pick two lines [math]l_1[/math] and [math]l_2[/math]. Create two lines through lines [math]l_1[/math] and [math]l_2[/math] intersecting at point [math]P[/math] which is not on either of the original lines. This is based on Projective Axiom 4 which states that all points will not lie on one line and Axiom 3 which states that two lines have at least one point in common. For every point [math]X[/math] on [math]l_1[/math], the line [math]\overline{PX}[/math] intersects [math]l_2[/math] because Proective Axiom 3 says that two lines have at least one point in common. This creates the one-to-one correspondence because each point on line [math]l_1[/math] through point [math]P[/math] intersects line [math]l_2[/math]. [math]\diamondsuit[/math]

Proof: Pick two points [math]P[/math] and [math]Q[/math]. Create a line through each of the points and notice that they have a point [math]R[/math] in common. Based on Projective Axiom 3, any two distinct lines have at least one point in common. Therefore, any lines through points [math]P[/math] and [math]Q[/math] would have a point in common. From this we can conclude, that there is a one-to-one correspondence with the lines through each of the points because they each share one point. [math]\diamondsuit[/math]