  Suppose $N \subset M$ is an inclusion of $II_1$-factors of finite index. If
$N$ can be generated by a finite set of elements, then there exist finite
generating sets $X$ for $N$ and $Y$ for $M$ such that $\delta_0(X) \geq
\delta_0(Y)$, where $\delta_0$ denotes Voiculescu's microstates (modified) free
entropy dimension. Moreover given $\epsilon >0$ one has $\delta_0(F) \geq
\delta_0(G) \geq ([M:N]^{-2} -\epsilon) \cdot (\delta_0(F) -1) + 1 - \epsilon$
for certain generating sets $F$ for $N$ and $G$ for $M$.
