Definition same_side' := fun A B X Y =>
  X <> Y /\ forall l, IncidentL X l -> IncidentL Y l -> same_side A B l.

Lemma axiom_hcong_4_existence : forall A B C O X P, ~ Col_H P O X -> ~ Col_H A B C -> exists Y, CongA A B C X O Y (* /\ ~Col O X Y *) /\ same_side' P Y O X.