Proof with Geometry.
unfold on_inter_parallel_parallel.
intros.
DecompAndAll.
assert (P<>Q).
unfold not;intro;subst P...
assert (U<>V).
unfold not;intro;subst U...
eapply elim_length_ratio_on_parallel_d_2...
apply on_parallel_to_on_parallel_d...
unfold on_parallel;repeat split...
unfold not;intro.
assert (parallel A Y Y R).
eapply parallel_transitivity.
apply H.
Geometry.
Geometry.
assert (Col Y A R).
assert (parallel Y A Y R)...
eapply par_col_col_1.
apply H12.
Geometry.
assert (Col Y R W).
eapply col_trans_1 with (B:= A)...
assert (parallel U V Y R).
eapply col_par_par.
apply H3.
Geometry.
Geometry.
assert (Y<>R).
unfold not;intro;subst Y.
clear H11 H12 H13 H14 H7.
assert (parallel U V R A).
eapply col_par_par.
apply H3.
Geometry.
Geometry.
assert (parallel U V P Q).
eapply parallel_transitivity.
apply H1.
Geometry.
Geometry.
intuition.
assert (parallel U V P Q).
eapply parallel_transitivity.
apply H15.
Geometry.
Geometry.
intuition.
Qed.

Theorem aux_co_side_1 :
forall A B P Q M : Point,
Q <> M ->
inter_ll M A B P Q ->
S Q A B <> 0.