Proof 7.4
Prove Theorem 7.5, that an affine geometry of order has lines.
Proof: Consider an affine plane of order k. From Theorem 7.2, we know that each point in this affine plane lies on lines. We know this to be true because Axiom 3 tells us that any point not on line lies on exactly one line parallel to . We also know that the other lines pass through this point. Therefore, Theorem 7.2 holds true. From Theorem 7.3, we know that an affine plane of order k has points. Previously, we showed that there are lines. On each of the lines, there are points and other points and each of the points that are not P will lie on exactly one line, so the total number points can be found using . Therefore, Theorem 7.3 holds true. Since there are points and lines through these points. Using these two facts, we can see that there are lines because each point has lines that pass through it. This would give us lines in the geometry. However, lines are repeated because a line must lie on two points by Axiom 1. To account for this repetition we need to divide the total number of lines by this repetition. Following through with the algebra, we see that there are lines in an affine geometry of order k.