  We investigate the requirement of the existence of two degenerate vacua of
the effective potential as a function of the Weinberg-Salam Higgs scalar field
norm, as suggested by the multiple point principle, in an extension of the
Standard Model including see-saw scale physics. Results are presented from an
investigation of an extension of the Standard Model to the gauge symmetry group
SU(3)_C\times SU(2)_L\times U(1)'\times \tilde U(1), where two groups U(1)' and
\tilde U(1) originate at the see-saw scale M_{SS}, when heavy (right-handed)
neutrinos appear. The consequent unification of the group SU(3)_C\times
SU(2)_L\times U(1)' into the flipped SU(5) at the GUT scale leads to the group
SU(5)\times \tilde U(1). We assume the position of the second minimum of the
effective potential coincides with the fundamental scale, here taken to be the
GUT scale. We solve the renormalization group equations in the one-loop
approximation and obtain a top-quark mass of 171\pm 3 GeV and a Higgs mass of
129\pm 4 GeV, in the case when the Yukawa couplings of the neutrinos are less
than half that of the top quark at the GUT scale.
