  It is shown that an interacting theory, defined on a regular lattice, must
have a vector-like spectrum if the following conditions are satisfied:
(a)~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.
