Proof with Geometry.
intros.
unfold S4 in |- *.
assert (S P A Q = - S A P Q)...
assert (S P Q B = S B P Q)...
assert (S P D Q = - S D P Q)...
assert (S P Q C = S C P Q)...
rewrite H0.
rewrite H1.
rewrite H2.
rewrite H3.
assert (T := l2_11a_strong_strong_strong A B C D P Q H)...
RewriteVar (S A P Q) T...
ring...
Qed.

Theorem l2_12a :
forall A B C D P : Point,
parallelogram A B C D ->
S P A B = S P D C - S A D C.