  On the example of a lattice-gas model, a convincing confirmation is obtained
for the direct relationship between the condensation phenomenon and divergent
behavior of the virial expansions for pressure and density in powers of
activity. The present study analytically proves the pressure equality for the
low-density and high-density virial expansions in powers of density (in terms
of irreducible cluster integrals or virial coefficients) exactly at the
symmetrical points, where their isothermal bulk modulus vanishes, as well as
for the corresponding expansions in powers of activity (in terms of reducible
cluster integrals) at the same points (the points of their divergence). For
lattice-gas models of arbitrary geometry and dimensions, a simple and general
expression is derived for the phase-transition activity (the convergence radius
of activity expansions) that, in particular, exactly matches the well-known
phase-transition activity of the Lee -- Yang model. In addition, the study
demonstrates that Mayer's expansion with the constant (volume independent)
cluster integrals remains correct up to the condensation beginning, and the
actual density-dependence may be taken into account for the high-order
integrals only in more dense regimes beyond the saturation point.
