Definition cut' A B X Y := X <> Y /\ forall l, IncidL X l -> IncidL Y l -> cut l A B.Global Instance cut_morphism : Proper(EqL ==> eq ==> eq ==> iff) cut.Global Instance same_side_morphism : Proper(eq ==> eq ==> EqL ==> iff) same_side.

Lemma congaH_existence_congH : forall A B C O X P U V : Point, U<>V -> ~ ColH P O X -> ~ ColH A B C -> exists Y : Point, CongaH A B C X O Y /\ same_side' P Y O X /\ CongH O Y U V.