Proof Theorem 7.7
In a projective plane, any two distinct lines have exactly one point in common.
Proof:
Consider two distinct lines l and l'. Projective Axiom 3 tells us that l and l' have at least one point, Q, in common. Now suppose there exists a second point, P, distinct from point Q and shared by l and l'. That is, we have two distinct points, P and Q, that have two lines, l and l', in common. This contradicts Projective Axiom 2. Thus, l and l' cannot have a second point in common. Therefore, l and l' only have one point in common. We can then conclude that any two distinct lines in a projective plane have exactly one point in common.