Multidimensional Kruskal-Katona theorem

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.