unfold Same_dir.
right.
exists B.
split.
unfold Out.
repeat split.
intro.
subst C.
apply between_identity in H1.
tauto.
auto.
right.
assumption.
apply eqv_refl.
Qed.

Lemma bet_same_dir2 : forall A B C, A <> B -> B <> C -> Bet A B C -> Same_dir A B B C.
intros.