Betweenness Proposition I
Proof: Consider the four points A,B,C,D. Assume A*B*C and B*C*D. By Proposition of line separation, A*B*C implies A and C are on opposite sides of the point B. Similarly, B*C*D implies that C and D are on the same side of B. Thus, because C is on the opposite side of B from A and C is on the same side of B as D, we know that A is on the opposite side of B as D. That is, A*B*D. Again, by Proposition of line separation, A*B*C implies that A and B are on the same side of C. Similarly, B*C*D implies that B and D are on opposite sides of C. Thus, as before, we can infer that A and D are on opposite sides of C. That is, A*C*D. Now assume A*B*D and B*C*D. By the Proposition of line separation, A*B*D implies that A and D are on opposite sides of B and B*C*D implies that C and D are on the same side of B. Thus, A and C must be on opposite sides of B. That is, A*B*C. Similarly, A*B*D implies that A and B are on the same side of D and B*C*D implies that B and C are on the same side of D. Thus, A and C are on the same side of D. That is, A*C*D.