Proof: Theorem 7.7
Proof 7.7
Theorem 7.7: In a projective plane, any two distinct lines have exactly one point in common.
Proof (By Contradiction):
Case 1: Assume two distinct lines, and , exist and intersect at points, and . Note that from Projective Axiom 2, it is known that and share exactly one line. However, this contradicts the assumption that two distinct lines, and , exist simultaneously. Therefore, we know that two distinct lines have exactly one point in common.
Case 2: Assume we have two distinct lines, and , which share no points. This can never be the case because we would contradict Projective Axiom 3 which states that any two distinct lines have at least one point in common.
Therefore, in a projective plane, any two distinct lines have exactly one point in common.