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 plane_separation : forall A B X Y p, ~ ColH A X Y -> ~ ColH B X Y -> IncidP A p -> IncidP B p -> IncidP X p -> IncidP Y p -> cut' A B X Y \/ same_side' A B X Y.