unfold Projp in H.
spliter.
induction H0.
spliter.
induction(eq_dec_points P' A).
subst P'.
right.
apply perp_not_eq_1 in H1.
assumption.
left.
assumption.
spliter.
subst P'.
induction(eq_dec_points P A).
subst P.
right.
assumption.
left.
assumption.
Qed.

(* Specifications: *)
Lemma projp_is_project :
forall P P' A B,
Projp P P' A B ->
exists X, exists Y, Proj P P' A B X Y.
Lemma projp_is_project_perp :
forall P P' A B,
Projp P P' A B ->
exists X, exists Y, Proj P P' A B X Y /\ Perp A B X Y.
Lemma projp_to_project :
forall P P' A B X Y,
Perp A B X Y ->
Projp P P' A B ->
Proj P P' A B X Y.
Lemma project_to_projp :
forall P P' A B X Y,
Proj P P' A B X Y ->
Perp A B X Y ->
Projp P P' A B.
Lemma projp_project_to_perp :
forall P P' A B X Y,
P <> P' ->
Projp P P' A B ->
Proj P P' A B X Y ->
Perp A B X Y.
Lemma project_par_project :
forall P P' A B X Y X' Y',
Proj P P' A B X Y ->
Par X Y X' Y' ->
Proj P P' A B X' Y'.
Lemma project_project_par :
forall P P' A B X Y X' Y',
P <> P' ->
Proj P P' A B X Y ->
Proj P P' A B X' Y' ->
Par X Y X' Y'.
Lemma projp_id : forall P P' Q' A B, Projp P P' A B -> Projp P Q' A B -> P' = Q'.
intros.