= Solution
Take the <four-divergence> of the <Proca equation>. Commuting partial derivatives and using that $F^{ab}$ is an <antisymmetric second-rank tensor> give
$$
\partial_b\partial_aF^{ab}
=\tfrac12\partial_a\partial_b(F^{ab}+F^{ba})=0,
\qquad m^2\partial_bA^b=0.
$$
Consequently, \b[for $m\ne0$, $\partial_aA^a=0$]. This is the <Lorenz constraint in Proca theory>: an <equation of motion> enforces it, rather than a choice of <gauge fixing>. It leaves $(\Box-m^2)A^a=0$. At $m=0$ the divergence argument supplies no such constraint; the massless <gauge symmetry> requires a separate treatment.
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