Lemma hilbert_s_version_of_pasch_aux : forall A B C I P, Coplanar A B C P -> ~ Col A I P -> ~ Col B C P -> Bet B I C -> B <> I -> I <> C -> B <> C -> exists X, Col I P X /\ ((Bet A X B /\ A <> X /\ X <> B /\ A <> B) \/ (Bet A X C /\ A <> X /\ X <> C /\ A <> C)).