  The long-standing belief is that the mean-field-like decoupling procedures
applied to the slave-particle representations of the problems with strong local
interaction violate Luttinger sum rule. The number of occupied resonant states
is small and typically equal to the deviation from the sum rule, shedding doubt
on the overall results. It is therefore illustrated, on the example of the
Emery model for the high-Tc superconductors, that, through the consistent
application of the mean-field procedure to the Hamiltonian and the propagators,
the sum rule is restored and the resonant band conserved. In addition to the
resonant band, the electron spectrum contains large number of occupied states
at the bare site-energy of the ion with strong repulsion. These results can be
straightforwardly generalized to the other similar problems.
