Lemma legendre_aux1 : greenberg_s_axiom -> triangle_postulate -> forall A1 A2 B1 B2 C1 C2 P, Perp2 A1 A2 B1 B2 P -> ~ Col A1 A2 P -> Col P B1 B2 -> Coplanar A1 A2 B1 B2 -> Par A1 A2 C1 C2 -> Col P C1 C2 -> ~ TS B1 B2 A1 C1 -> Col C1 B1 B2.