Continuation of separate analytical functions with fine features in each section

Let $D\subset \mathbb{C}$ and $G\subset \mathbb{C}$ be bounded simply connected domains, $E\subset D$ and $F\subset G$ be regular compact sets. If $f(z,w)$ is a separate analytic function, with finite number of singular points in each section $D\times \{w\}$ and $\{z\}\times G$, $\forall (z, w)\in E\times F$, on the set
$$ X=(D\times F)\bigcup (E\times G), $$
then it continues holomorphically into the domain
$$ \hat{X} = \{(z,w)\in D\times G: {{\omega }^{*}}(z,E,D)+{{\omega }^{*}} (w,F,G)<1 \}, $$
except for some analytic set $S$. Here ${{\omega }^{*}}(z,E,D)$ is the harmonic measure of the set $E\subset D\subset \mathbb{C}$ with respect to the domain $D$.