assert(Col A B C').
eapply (proj3_col A B C A B C' P Q); auto.
assert(HH1:= H1).
assert(HH2:=H2).
assert(HH3:=H3).
unfold proj in HH1.
unfold proj in HH2.
unfold proj in HH3.
spliter.
induction H10; induction H8; induction H6; spliter.
apply perp_not_eq_2 in H15.
tauto.
apply perp_not_eq_2 in H14.
tauto.
apply perp_not_eq_2 in H14.
tauto.
apply perp_not_eq_2 in H13.
tauto.
apply perp_not_eq_2 in H13.
tauto.
apply perp_not_eq_2 in H12.
tauto.
assert(Col P A B).
ColR.
assert(Col Q A B).
ColR.
assert(Col P Q C).
eapply (col3 A B); Col;
eapply (l8_14_2_1b_bis _ _ _ _ C') in H7.
contradiction.
subst C.
reflexivity.
Qed.

(* Specifications: *)
Lemma proj_inv_exists : forall P Q A', P <> Q -> Col P Q A'  -> exists A, A <> A' /\ proj A P Q A'.
intros.