unfold disjoint.
intro.
destruct H0 as [P []].
assert(HP:= betH_trans0 A P B C H0 H).
spliter.
apply between_only_one' in H1.
spliter.
contradiction.
Qed.

Lemma outH_expand: forall A B C,
outH A B C -> outH A B C  /\ ColH A B C /\ A<>C /\ A<>B.