Solution Exercise 4.13.
Definition of Term Commonly Used in Mathematics:
Trivial – If I show you how to do this, you are in the wrong class.

Exercise 4.13. Prove the existence of a complete quadrilateral.

A short proof would be to apply the principle of duality to Exercise 4.5. A direct proof follows below.

Proof. By the Dual of Axiom 3, there exist at least four lines a, b, c, and d, no three of which are concurrent. By the Dual of Axiom 1, the points a · b, a · c, a · d, b · c, b · d,  and c · d exist. The six points are distinct since if two of the points were the same point, then three of the lines a, b, c, or d would be concurrent, which is a contradiction. Therefore, the four lines a, b, c, and d determine a quadrilateral abcd.//