unfold proj in *.
spliter.
induction H4; induction H3; induction H2; spliter.
eapply (col3 P Q); Col.
subst C'.
eapply (col3 P Q); Col.
subst B'.
eapply (col3 P Q); Col.
subst C'.
subst B'.
eapply (col3 P Q); Col.
subst A'.
eapply (col3 P Q); Col.
subst C'.
subst A'.
eapply (col3 P Q); Col.
subst A'.
subst B'.
eapply (col3 P Q); Col.
subst A'.
subst B'.
subst C'.
eapply (col3 P Q); Col.
Qed.

Lemma proj3_id : forall A B C C' P Q, A <> B -> Col A B C -> proj A P Q A -> proj B P Q B -> proj C P Q C' -> C = C'.
intros.