  We present a generalization of a version of the Kruskal-Katona theorem due to
Lovasz. A shadow of a d-tuple (S_1,...,S_d) in binom{X}{r}^d consists of
d-tuples (S_1',...,S_d') in binom{X}{r-1}^d obtained by removing one element
from each of S_i. We show that if a family F in binom{X}{r}^d has size
|F|=binom{x}{r}^d for a real number x>=r, then the shadow of F has size at
least binom{x}{r-1}^d.
