Proof with Geometry.
intros.
cases_equality P D.
cases_equality Q C.
cut (weak_2_parallelogram P Q D C).
apply weak_2_parallelogram_weak_3_parallelogram.
apply parallel_side_eq_weak_weak_para...
intuition.
subst P.
unfold weak_3_parallelogram.
exists D.
unfold mid_point.
intuition.
unfold parallel, S4 in H.
basic_simpl...
cases_equality Q C.
subst Q.
unfold weak_3_parallelogram.
exists C.
unfold mid_point.
intuition.
unfold parallel, S4 in H.
basic_simpl...
cut (weak_2_parallelogram P Q D C).
apply weak_2_parallelogram_weak_3_parallelogram.
apply parallel_side_eq_weak_weak_para...
intuition.
Qed.

Lemma on_line_dex_spec_strong_f :
forall P Q C D : Point,
parallel P Q C D -> P<>Q ->
exists Y : Point, Col Y P Q /\ P ** Y = C ** D /\ weak_3_parallelogram P Y D C.