E-solid locally inverse semigroups

We prove that if S is an E-solid locally inverse semigroup, and rho is an inverse semigroup congruence on S such that the idempotent classes of rho are completely simple semigroups then S is embeddable into a lambda-semidirect product of a completely simple semigroup by S/rho. Consequently, the E-solid locally inverse semigroups turn out to be, up to isomorphism, the regular subsemigroups of lambda-semidirect products of completely simple semigroups by inverse semigroups.