Solution Basic Theorems Exercise 4.2.
Prove Theorem 4.2.
If you don't learn from your mistakes, there's no sense making them.
—Herbert V. Prochnow (1897–1998)

Important Note. You may not assume the existence of any points or any lines. You must prove that both a line and a point exist.

Theorem 4.2. There exist a point and a line that are not incident.

Proof.  By Axiom 3, there exist four points A, B, C, and D, no three of which are collinear. In particular A, B, and C are not on the same line. By Axiom 1, there exists a line l such that B and C are incident with l. Hence A and l are a point and line that are not incident. Therefore, there exist a point and a line that are not incident. //