  Two families $\mathcal{F},\mathcal{G}$ of $k$-subsets of $\{1,2,\ldots,n\}$
are called non-trivial cross $t$-intersecting if $|F\cap G|\geq t$ for all
$F\in \mathcal{F}, G\in \mathcal{G}$ and $|\cap \{F\colon F\in
\mathcal{F}\}|<t$, $|\cap \{G\colon G\in\mathcal{G}\}|<t$. In the present
paper, we determine the maximum product of the sizes of two non-trivial cross
$t$-intersecting families of $k$-subsets of $\{1,2,\ldots,n\}$ for $n\geq
4(t+2)^2k^2$, $k\geq 5$, which is a product version of the Hilton-Milner-Frankl
Theorem.
