  Given an arbitrary finite dimensional Hamiltonian H_0, we consider the model
H=H_0+\Delta H, where \Delta H is a generic fully connected interaction. By
using the strong law of large numbers we easily prove that, for all such
models, a generalized Curie-Weiss mean-field equation holds. Unlike traditional
mean-field models the term H_0 gives rise to short-range correlations and,
furthermore, when H_0 has negative couplings, first-order phase transitions and
inverse transition phenomena may take place even when only two-body
interactions are present. The dependence from a non uniform external field and
finite size effects are also explicitly calculated. Partially, these results
were derived long ago by using min-max principles but remained almost unknown.
