[b][u]Lemma 7.1:[/u][/b] 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)[br][br]Proof:[br][br]Pick two lines, [math]l_1[/math] and [math]l_2[/math]. Using points a and c on [math]l_1[/math], and points b and d on [math]l_2[/math], create lines (a,b) and (c,d) which intersect at point P, where P is not on either [math]l_1[/math] or [math]l_2[/math]. 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 [math]l_1[/math], the line (P, X) intersects [math]l_2[/math]. Therefore, any line incident on any point of [math]l_1[/math] and P intersects [math]l_2[/math]. Analogously, for any point Y on [math]l_2[/math], the line (P, Y) intersects [math]l_1[/math]. [br][br]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. [br][br][b][u]State the Dual of Lemma 7.1:[/u] [/b]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)[br][br]Proof:[br][br]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, [math]l_3[/math] through points X,W and [math]l_4[/math] through points Y,Z. By Dual Axiom 2, we know that [math]l_3[/math] and [math]l_4[/math] must have a point, P, in common. By Dual Axiom 3, the following lines must exist:[br][br][math]l_5[/math] through W,Y[br][math]l_6[/math] through Z,Y[br][math]l_7[/math] through Z,X[br][math]l_8[/math] through X,W.[br][br]By Dual Axiom 2, for any line, [math]l_n[/math], we know that [math]l_n[/math] 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.