induction (eq_dec_points A B).
subst B.
apply same_dir_null in H.
subst D.
left.
tauto.
unfold Same_dir in *.
induction H.
left.
tauto.
ex_and H D'.
right.
assert(HH:=vector_construction C D A).
ex_and HH B'.
exists B'.
split.
unfold EqV in H1.
unfold EqV in H2.
unfold Out in *.
spliter.
induction H1; induction H2.
repeat split.
auto.
intro.
subst B'.
induction H2.
apply H2.
Col.
apply plgf_sym in H2.
apply plgf_trivial_neq in H2.
spliter.
auto.
induction H4.
right.
apply (plg_plg_bet C D _ _ D').
apply H2.
apply H4.
apply plg_sym.
apply plg_comm2.
assumption.
left.
apply (plg_plg_bet C D' _ _ D).
apply plg_sym.
apply plg_comm2.
apply H1.
apply H4.
assumption.
spliter.
subst D.
tauto.
spliter.
subst B.
tauto.
spliter.
subst D.
tauto.
assumption.
Qed.

Lemma same_dir_trans : forall A B C D E F, Same_dir A B C D -> Same_dir C D E F -> Same_dir A B E F.
intros.