Proof with Tarski.
intros.
cases_col A B C.
assert (exists C', ~ Col A B C').
eapply l6_25...
DecompEx H1 C'.
assert (  exists P : Point,
(exists T : Point, Perp A B P A /\ Col A B T /\ Bet C' T P)).
apply (l8_21_aux A B C');auto.
DecompEx H1 P.
DecompEx H3 T.
exists P.
exists C.
intuition.
eapply l8_21_aux;auto.
Qed.

Lemma perp_in_commutativity : forall A B C D X,
Perp_in X A B C D -> Perp_in X B A D C.