Proof with Geometry.
intros...
cases_equality A M...
assert (Col A M B)...
assert (Col Q M P)...
rewrite <- H2 in H5...
assert (Col A B P)...
eauto with Geom...
assert (~ Col P A Q \/ ~ Col P Q B)...
apply par_not_all_col...
case H7; intro...
assert (Col P A A)...
rewrite <- H2 in H8...
intuition...
rewrite H2 in H6...
assert (Col P Q B)...
intuition...
Qed.

Theorem inter_unicity_2 :
forall A B P Q M N: Point,
~ parallel P Q A B ->
Col A B M -> Col P Q M ->
Col A B N -> Col P Q N ->
M = N.