Problem of the identical vanishing of Euler-Lagrange derivatives in field theory

I prove that the necessary and sufficient condition for two Lagrangian densities L1(A;A,) and L2(A;A,) to have exactly the same Euler-Lagrange derivatives is that their difference (A;A,) be the divergence of (A;A,;) with a given dependence on A,. The main point is that depends on A, but does not depend on second derivatives of the field A. Therefore, the function need not be linear in A,.