Lemma strong_parallel_postulate_implies_tarski_s_euclid_aux : strong_parallel_postulate -> (forall A B C D T, A <> B -> A <> C -> A <> D -> A <> T -> B <> C -> B <> D -> B <> T -> C <> D -> C <> T -> D <> T -> ~ Col A B C -> Bet A D T -> Bet B D C -> exists B', exists B'', exists MB, exists X, Bet A B X /\ Par_strict B C T X /\ BetS B MB T /\ BetS B' MB B'' /\ Cong B MB T MB /\ Cong B' MB B'' MB /\ Col B B' D /\ Bet B'' T X /\ B <> B' /\ B'' <> T).