  We consider interacting theories with a compact internal symmetry group on a
regular lattice. We show that the spectrum is necessarily vector-like provided
the following conditions are satisfied: (a)~weak form of locality,
(b)~relativistic continuum limit without massless bosons, and (c)~pole-free
effective vertex functions for conserved currents.
  The proof exploits the zero frequency inverse retarded propagator of an
appropriate set of interpolating fields as an effective quadratic hamiltonian,
to which the Nielsen-Ninomiya theorem is applied.
  The main results of this paper have been reported in WIS-93/56-JUNE-PH,
hep-lat/9306023.
