  We study the finite temperature symmetry behaviour of O(N_1) \times O(N_2)
scalar models on the lattice and we prove that at sufficiently high
temperatures and in arbitrary dimensions their full symmetry is always restored
or, equivalently, that the phenomenon of Symmetry Non Restoration which,
according to lowest order perturbation theory, takes place in the continuum
version of these models, does not occur on the lattice.
