Theorem 7.7: In a projective plane, any two distinct lines have exactly one point in common. [br][br]Proof (By Contradiction): [br][br]Case 1: Assume two distinct lines, [math]l_1[/math] and [math]l_2[/math], exist and intersect at points, [math]x[/math] and [math]y[/math]. Note that from Projective Axiom 2, it is known that [math]x[/math] and [math]y[/math] share exactly one line. However, this contradicts the assumption that two distinct lines, [math]l_1[/math] and [math]l_2[/math], exist simultaneously. Therefore, we know that two distinct lines have exactly one point in common. [br][br]Case 2: Assume we have two distinct lines, [math]l_1[/math] and [math]l_2[/math], 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.[br][br]Therefore, in a projective plane, any two distinct lines have exactly one point in common.