If A, B, and C are three non-collinear points and X is a point of AC, then X is the only point on both AC and BX.
I. Given non-collinear points A, B, and C, with a point, X, on AC, assume that there is also a point, Y, on both AC and BX. [In other words, we're going to hypothesize that Theorem 2 is false and show that leads to a contradiction].
II. By Axiom III, points X and Y are contained in only one line, contradicting the hypothesis that the points X and Y are contained in both AC and BX.
III. Since the hypothesis is false, Theorem 2 must be true.