Let \(D_j\subset\mathbb{C}^{k_j}\) be a pseudoconvex domain and \(A_j \subset D_j\) be a locally pluriregular set, \(j=1,\dots,N\), \(N\geq 2\). Put \[ X:= \bigcup^N_{j=1} A_1\times\cdots \times A_{j-1}\times D_j\times A_{j+1}\times \cdots\times A_N. \] The main result of this paper is the following.
Theorem. Let \(U\) be an open connected neighborhood of \(X\) and \(M\nsubseteq U\) be an analytic set. Then there exists a pure one-codimensional analytic subset \(\widehat M\) of the “envelope of holomorphy” \(\widehat X\) of \(X\) such that for every separately holomorphic function \(f\in O_s(X\setminus M)\) there exists only one function \(\widehat f\in O(\widehat X\setminus\widehat M)\) with \(\widehat f|_{X \setminus M}=f\).