Proof: Lemma 7.1
Lemma 7.1
Proof: Lemma 7.1 and Dual of Lemma 7.1
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. (SEEN ABOVE)
Proof:
Pick two lines, and . Using points a and c on , and points b and d on , create lines (a,b) and (c,d) which intersect at point P, where P is not on either or . Note that this is possible by Projective Axiom 4 because it states that there is a set of four distinct points, a,b,c,d, where no three are collinear. Also using Projective Axiom 3, we know that (a,b) and (c,d) have at least one point in common - in this case, point P. Note that by Projective Axiom 3, for any point X on , the line (P, X) intersects . Therefore, any line incident on any point of and P intersects . Analogously, for any point Y on , the line (P, Y) intersects .
Therefore, in a projective plane, the points on one line can be put into one-to-one correspondence with the points on any other line.
State the Dual of Lemma 7.1: In a projective plane, the lines on one point can be put into one-to-one correspondence with the lines on any other point. (SEEN BELOW)
Proof:
Pick two points, X and Y. By Projective Axiom 4, note that points W and Z must exist such that no three of X, Y, W, Z are collinear. Create two distinct lines, through points X,W and through points Y,Z. By Dual Axiom 2, we know that and must have a point, P, in common. By Dual Axiom 3, the following lines must exist:
through W,Y
through Z,Y
through Z,X
through X,W.
By Dual Axiom 2, for any line, , we know that must have exactly one point in common with every other line. Therefore, the lines on one point can be put into one-to-one correspondence with the lines on any other point.