  We study a lattice Nambu-Jona-Lasinio model with interacting staggered
fermions in the Kogut-Susskind Hamiltonian formalism. The model has a discrete
chiral symmetry but not the usual continuous chiral symmetry. In a strong
coupling regime for the four-fermion interaction, we prove that the mass of the
fermions is spontaneously generated at sufficiently low non-zero temperatures
in the dimensions $\nu \ge 3$ of the model and zero temperature in $\nu \ge 2$.
Due to the phase transition, the discrete chiral symmetry of the model is
broken. Our analysis is based on the reflection positivity for fermions and the
method of the infrared bound.
