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

I prove that the necessary and sufficient condition for two Lagrangian densities ${\mathcal{L}}_1({\psi }^A;{{\psi }^A}_{,\alpha })$ and ${\mathcal{L}}_2({\psi }^A;{{\psi }^A}_{,\alpha })$ to have exactly the same Euler-Lagrange derivatives is that their difference $\Delta ({\psi }^A;{{\psi }^A}_{,\alpha })$ be the divergence of ${\omega }^{\mu }({\psi }^A;{{\psi }^A}_{,\alpha };x^{\mu })$ with a given dependence on ${{\psi }^A}_{,\alpha }$. The main point is that ${\omega }^{\mu }$ depends on ${{\psi }^A}_{,\alpha }$ but $\Delta$ does not depend on second derivatives of the field ${\psi }^A$. Therefore, the function $\Delta$ need not be linear in ${{\psi }^A}_{,\alpha }$.