  Let $L$ be any finite distributive lattice and $B$ be any boolean predicate
defined on $L$ such that the set of elements satisfying $B$ is a sublattice of
$L$. Consider any subset $M$ of $L$ of size $k$ of elements of $L$ that satisfy
$B$. Then, we show that $k$ generalized median elements generated from $M$ also
satisfy $B$. We call this result generalized median theorem on finite
distributive lattices. When this result is applied to the stable matching, we
get Teo and Sethuraman's median stable matching theorem. Our proof is much
simpler than that of Teo and Sethuraman. When the generalized median theorem is
applied to the assignment problem, we get an analogous result for market
clearing price vectors.
