Verbal subgroups of complete direct products of groups

It is proved that if $V(X)$ is a proper verbal subgroup of a free group $X$ of countable rank, then a verbal subgroup $V(H)$ of the complete direct product $H=\widetilde\Pi^\times X_i$ of a countable number of isomorphic copies $X_i$ of $X$ differs from the complete direct product $\widetilde\Pi^\times V(X_i)$.