Proof with Geometry.
intros.
assert (S B D C = S A D C).
assert (S A D C + S C D C = S B D C + S D D C).
apply l2_11a...
basic_simpl...
assert (S4 D B C P = S D B C + S D C P)...
assert (S4 D B C P = S P D C - S A D C).
rewrite H1.
uniformize_signed_areas.
rewrite H0.
ring.
assert (S P A B = S P D B - S P C B).
assert (S A B P + S C B P = S D B P + S B B P).
apply l2_11a...
basic_simpl.
uniformize_signed_areas.
rewrite <- H3.
ring.
rewrite H3.
assert (S4 D B C P = S D B P - S C B P)...
uniformize_signed_areas.
congruence.
Qed.

Theorem l2_12a_strong_3 :
forall A B C D P : Point,
weak_3_parallelogram A B C D ->
S P A B = S P D C - S A D C.