induction(col_dec A B T).
exists T.
unfold proj.
split.
assumption.
right.
split;
auto.
assert(HH:=l8_18_existence A B T H0).
ex_and HH P.
exists P.
unfold proj.
split.
assumption.
left.
repeat split;auto.
Qed.

Lemma proj_per : forall A B T P, A <> B -> proj T A B P -> Per T P A /\ Per T P B /\ Col A B P.
intros.