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 A B C D P Q H)...
RewriteVar (S A P Q) T...
ring...
Qed.

Theorem l2_11b_strong_strong :
forall A B C D P Q : Point, weak_2_parallelogram A B C D -> S4 P A Q B = S4 P D Q C.