unfold Same_dir in H.
induction H.
tauto.
ex_and H D.
apply null_vector in H0.
subst D.
unfold Out in H.
tauto.
Qed.

(* Specifications: *)
Lemma plgs_out_plgs :
forall A B C D B' C',
Parallelogram_strict A B C D ->
Out A B B' ->
Out D C C' ->
Cong A B' D C' ->
Parallelogram_strict A B' C' D.
Lemma plgs_plgs_bet :
forall A B C D B' C',
Parallelogram_strict A B C D ->
Bet A B B' ->
Parallelogram_strict A B' C' D ->
Bet D C C'.
Lemma plgf_plgf_bet :
forall A B C D B' C',
Parallelogram_flat A B C D ->
Bet A B B' ->
Parallelogram_flat A B' C' D ->
Bet D C C'.
Lemma plg_plg_bet :
forall A B C D B' C',
Parallelogram A B C D ->
Bet A B B' ->
Parallelogram A B' C' D ->
Bet D C C'.
Lemma plgf_out_plgf :
forall A B C D B' C',
Parallelogram_flat A B C D ->
Out A B B' ->
Out D C C' ->
Cong A B' D C' ->
Parallelogram_flat A B' C' D.
Lemma plg_out_plg :
forall A B C D B' C',
Parallelogram A B C D ->
Out A B B' ->
Out D C C' ->
Cong A B' D C' ->
Parallelogram A B' C' D.
Lemma same_dir_sym : forall A B C D, Same_dir A B C D -> Same_dir C D A B.
intros.