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

(* Specifications: *)
Lemma same_dir_ts :
forall A B C D,
Same_dir A B C D ->
exists M, Bet A M D /\ Bet B M C.
Lemma one_side_col_out :
forall A B X Y,
Col A X Y ->
OS A B X Y ->
Out A X Y.
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.
Lemma same_dir_out : forall A B C, Same_dir A B A C -> Out A B C \/ A = B /\ A = C.
intros.