Let $D, G\subset\mathbb C$ be domains, let $A\subset D$, $B\subset G$ be locally regular sets, and let $X:=(D\times B)\cup(A\times G)$. Assume that $A$ is a Borel set. Let $M$ be a proper analytic subset of an open neighborhood of $X$. Then there exists a pure $1$-dimensional analytic subset $\widehat M$ of the envelope of holomorphy $\skew3\widehat X$ of $X$ such that any function separately holomorphic on $X\setminus M$ extends to a holomorphic function on $\skew3\widehat X\setminus\widehat M$. The result generalizes special cases which were studied in [Ökt 1998], [Ökt 1999], and [Sic 2000].