  We reconsider the problem of the orbital dynamics of the innermost exoplanet
of the $\upsilon$-Andromedae system (i.e., $\upsilon$-And $\mathbf{b}$) into
the framework of a Secular Quasi-Periodic Restricted Hamiltonian model. This
means that we preassign the orbits of the planets that are expected to be the
biggest ones in that extrasolar system (namely, $\upsilon$-And $\mathbf{c}$ and
$\upsilon$-And $\mathbf{d}$). The Fourier decompositions of their secular
motions are injected in the equations describing the orbital dynamics of
$\upsilon$-And $\mathbf{b}$ under the gravitational effects exerted by those
two exoplanets. By a computer-assisted procedure, we prove the existence of KAM
tori corresponding to orbital motions that we consider to be very robust
configurations, according to the analysis and the numerical explorations made
in our previous article. The computer-assisted assisted proofs are successfully
performed for two variants of the Secular Quasi-Periodic Restricted Hamiltonian
model, which differs for what concerns the effects of the relativistic
corrections on the orbital motion of $\upsilon$-And $\mathbf{b}$, depending on
whether they are considered or not.
