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.

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.