Proof 7.11
a. Prove that the points on one line in a projective plane can be put in a one-to-one correspondence with the points on any other line (Lemma 7.1).
Proof: Pick two lines and . Create two lines through lines and intersecting at point 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 on , the line intersects 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 through point intersects line .
b. State the dual part of a. (Theorem 7.10)
Proof: Pick two points and . Create a line through each of the points and notice that they have a point in common. Based on Projective Axiom 3, any two distinct lines have at least one point in common. Therefore, any lines through points and 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.