  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.
