A remark on separate holomorphy

Let $X$ be a Riemann domain over ${\mathbb{C}}^k\times {\mathbb{C}}^{\mathrm{\ell }}$. 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 $a\mathrm{\notin }P$ is a domain of holomorphy with respect to the family $\{f{\mathrm{\mid }}_{X_a}:f\in \mathcal{F}\}$.