unfold Same_dir in H.
induction H.
tauto.
ex_and H C.
apply eqv_mid in H0.
unfold Midpoint in H0.
unfold Out in H.
spliter.
induction H3.
apply (between_equality _ _ C).
apply H0.
assumption.
apply False_ind.
apply H2.
apply (between_equality _ _ A).
Between.
assumption.
Qed.

Lemma opp_dir_id : forall A B, Opp_dir A B A B -> A = B.
intros.