unfold Same_dir in *.
induction H; induction H0; spliter.
left.
tauto.
ex_and H0 F'.
subst B.
subst D.
apply null_vector in H2.
subst F'.
unfold Out in H0.
tauto.
ex_and H D'.
subst D.
subst F.
unfold Out in H.
tauto.
ex_and H D'.
ex_and H0 F'.
right.
induction(eq_dec_points A B).
subst B.
apply null_vector in H1.
subst D'.
unfold Out in H.
tauto.
assert(HH:=vector_construction A B E).
ex_and HH F''.
exists F''.
split.
2: auto.
assert(C <> D /\ C <> D' /\ E <> F /\ E <> F').
unfold Out in *.
spliter.
repeat split;
auto.
spliter.
unfold EqV in *.
induction H1; induction H2; induction H4.
unfold Out in *.
spliter.
induction H10; induction H12.
repeat split.
auto.
intro.
subst F''.
induction H4.
apply H4.
Col.
apply plgf_sym in H4.
apply plgf_trivial_neq in H4.
tauto.
left.
assert(Bet E F' F'').
apply (plg_plg_bet C D _ _ D').
apply H2.
apply H12.
assert(Parallelogram C D' F'' E \/ C = D' /\ B = A /\ E = F'' /\ C = E).
apply(plg_pseudo_trans C D' B A E F'').
apply plg_sym.
apply plg_comm2.
auto.
apply plg_comm2.
auto.
induction H13.
2:tauto.
assumption.
apply (between_exchange4 _ _ F').
apply H10.
auto.
repeat split.
auto.
intro.
subst F''.
induction H4.
apply H4.
Col.
apply plgf_sym in H4.
apply plgf_trivial_neq in H4.
tauto.
assert(Bet E F'' F').
apply (plg_plg_bet C D' _ _ D).
3: apply H2.
2:apply H12.
assert(Parallelogram C D' F'' E \/ C = D' /\ B = A /\ E = F'' /\ C = E).
apply plg_pseudo_trans.
apply plg_sym.
apply plg_comm2.
auto.
apply plg_comm2.
auto.
induction H13.
assumption.
tauto.
apply (l5_3 _ _ _ F').
apply H10.
assumption.
repeat split.
auto.
intro.
subst F''.
induction H4.
apply H4.
Col.
apply plgf_sym in H4.
apply plgf_trivial_neq in H4.
tauto.
assert(Bet E F' F'').
apply (plg_plg_bet C D _ _ D').
apply H2.
apply H12.
assert(Parallelogram C D' F'' E \/ C = D' /\ B = A /\ E = F'' /\ C = E).
apply plg_pseudo_trans.
apply plg_sym.
apply plg_comm2.
auto.
apply plg_comm2.
auto.
induction H13.
assumption.
tauto.
apply (l5_1 _ F').
apply H8.
auto.
auto.
repeat split.
auto.
intro.
subst F''.
induction H4.
apply H4.
Col.
apply plgf_sym in H4.
apply plgf_trivial_neq in H4.
tauto.
assert(Bet E F'' F').
apply (plg_plg_bet C D' _ _ D).
3: apply H2.
2: apply H12.
assert(Parallelogram C D' F'' E \/ C = D' /\ B = A /\ E = F'' /\ C = E).
apply plg_pseudo_trans.
apply plg_sym.
apply plg_comm2.
auto.
apply plg_comm2.
auto.
induction H13.
auto.
tauto.
right.
apply (between_exchange4 _ _ F').
apply H13.
auto.
tauto.
tauto.
tauto.
tauto.
tauto.
tauto.
tauto.
Qed.

Lemma same_dir_comm : forall A B C D, Same_dir A B C D -> Same_dir B A D C.
intros.