Proof with finish.
intros.
unfold Proj in *.
spliter.
repeat split; auto.
intro.
apply H3.
eapply (par_col_par_2 _ C); Col; Par.
(*perm_apply (par_col_par X Y A C).*)
ColR.
Qed.

Lemma pj_col_project : forall P P' A B X Y,
A <> B -> X <> Y ->
Col P' A B ->
~ Par A B X Y ->
Pj X Y P P' ->
Proj P P' A B X Y.