  We prove that two Aleksandrov solutions of a generated prescribed Jacobian
equation have the same gradients at points where they are both differentiable.
For the optimal transportation case where two solutions can be translated to
agree at a point without changing the $g$-subdifferential at that point, we
recover the uniqueness up to a constant of solutions. For the general case, our
result is a new proof with less regularity assumptions of a key theorem
recently used to prove the uniqueness of solutions.
