unfold Same_dir.
induction (eq_dec_points A B).
left.
tauto.
right.
exists B.
split.
apply out_trivial.
auto.
apply eqv_refl.
Qed.

Lemma par_ts_same_dir :
forall A B C D, Par_strict A B C D ->
(exists M, Bet A M D /\ Bet B M C) ->
Same_dir A B C D.