  Let $X$ be a Riemann domain over $\mathbb C^k\times\mathbb C^\ell$. If $X$ is
domain of holomorphy with respect to a family $\mathcal F\subset\mathcal O(X)$,
then there exists a pluripolar set $P\subset\mathcal C^k$ such that every slice
$X_a$ of $X$ with $a\notin P$ is a domain of holomorphy with respect to the
family $\{f|_{X_a}: f\in\mathcal F\}$.
