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 axiom_five_segmentsH: forall A A' B B' C C' D D', A<>D -> A'<>D' -> B<>D -> B'<>D' -> C<>D -> C'<>D' -> CongH A B A' B' -> CongH B C B' C' -> CongH A D A' D' -> CongH B D B' D' -> BetH A B C -> BetH A' B' C' -> A' <> D' -> CongH C D C' D'.