  A notion of generalized $n$-semimodularity is introduced, which extends that
of (sub/super)mod\-ularity in four ways at once. The main result of this paper,
stating that every generalized $(n\colon\!2)$-semimodular function on the $n$th
Cartesian power of a distributive lattice is generalized $n$-semimodular, may
be considered a multi/infinite-dimensional analogue of the well-known Muirhead
lemma in the theory of Schur majorization. This result is also similar to a
discretized version of the well-known theorem due to Lorentz, which latter was
given only for additive-type functions. Illustrations of our main result are
presented for counts of combinations of faces of a polytope; one-sided
potentials; multiadditive forms, including multilinear ones -- in particular,
permanents of rectangular matrices and elementary symmetric functions; and
association inequalities for order statistics. Based on an extension of the FKG
inequality due to Rinott \& Saks and Aharoni \& Keich, applications to
correlation inequalities for order statistics are given as well.
