induction (same_dir_dec A B C D).
left.
assumption.
right.
unfold Opp_dir.
unfold Same_dir.
right.
assert(HH:= vector_construction A B D).
ex_and HH C'.
exists C'.
split.
2:auto.
unfold EqV in H1.
induction (eq_dec_points B C').
subst C'.
induction H1.
induction H1.
apply False_ind.
apply plgs_permut in H1.
apply plgs_sym in H1.
apply H1.
Col.
apply plgf_permut in H1.
apply plgf_trivial_neq in H1.
spliter.
subst D.
induction H.
apply False_ind.
apply H.
exists A.
split; Col.
spliter.
induction H4.
unfold Out.
repeat split; auto.
left.
Between.
induction H4.
unfold Out.
repeat split; auto.
apply False_ind.
apply H0.
apply same_dir_sym.
apply bet_same_dir2; auto.
unfold Out.
repeat split.
auto.
auto.
right.
assumption.
spliter.
subst A.
subst D.
apply par_distinct in H.
tauto.
induction H1.
assert(Col C' C D /\ Col D C D).
apply plg_par in H1.
spliter.
apply(parallel_uniqueness A B C D C' D D); Col.
apply par_distinct in H.
tauto.
assumption.
spliter.
clear H4.
unfold Out.
repeat split.
apply par_distinct in H.
tauto.
intro.
subst C'.
induction H1.
apply H1.
Col.
apply plgf_sym in H1.
apply plgf_trivial_neq in H1.
spliter.
apply par_distinct in H.
tauto.
induction H3.
left.
Between.
induction H3.
apply False_ind.
assert(Same_dir A B D C').
apply plg_opp_dir.
assumption.
assert(Same_dir C D D C').
apply bet_same_dir2.
apply par_distinct in H.
spliter.
auto.
intro.
subst C'.
induction H1.
apply H1.
Col.
apply plgf_sym in H1.
apply plgf_trivial_neq in H1.
spliter.
apply par_distinct in H.
tauto.
assumption.
apply False_ind.
apply H0.
apply (same_dir_trans _ _ D C').
apply H4.
apply same_dir_sym.
auto.
right.
assumption.
apply par_distinct in H.
tauto.
Qed.

Lemma same_dir_id : forall A B, Same_dir A B B A -> A = B.
intros.