assert(Same_dir C D D C).
apply (same_dir_trans _ _ A B).
apply same_dir_sym.
apply H.
assumption.
apply same_dir_id in H1.
subst D.
apply same_dir_sym in H.
apply same_dir_null in H.
subst B.
tauto.
Qed.

Lemma opp_dir_to_null : forall A B C D, Opp_dir A B C D -> Opp_dir A B D C -> A = B /\ C = D.
unfold Opp_dir.