  Let $K$ be a field, $D$ a finite distributive lattice and $P$ the set of all
join-irreducible elements of $D$. We show that if $\{y\in P\mid y\geq x\}$ is
pure for any $x\in P$, then the Hibi ring $\RRRRR_K(D)$ is level. Using this
result and the argument of sagbi basis theory, we show that the homogeneous
coordinate rings of Schubert subvarieties of Grassmannians are level.
