Formal proof Prove that every half plane is a nonempty setSo

Formal proof.

Prove that every half plane is a nonempty set.

Solution

You have an axiom which say that in the plane you have a point that is not on the line.

So, if that point lies in that half plane you want it to lie, you will prove that that part is nonempty.

Otherwise, you consider the line that pass through that point A and some point B on the line.

You will get a line through these points A and B. By axiom there exist a point C such that A-B-C, in that order. The point C must lie in your half plane.

Hence proved