  Following the work of Khare {\it et al}, we show that the generalization to
systems with spontaneous symmetry breaking of the Coleman-Hill theorem to
one-loop order, can be extended to the case including fermions with the most
general interactions. Although the correction to the parity-odd part of the
vacuum polarization looks complicated in the Higgs phase, it turns out that the
correction to the Chern-Simons term is identical to that in the symmetric
phase, with the difference coming only from the contribution of the would be
Chern-Simons term. We also discuss the implication of our result to nonabelian
systems.
