Extension of locally holomorphic mappings into a product of complex manifolds

It is proved that locally biholomorphic mappings from the punctured ball in $\mathbf C^n$ into a product of complex manifolds of positive dimension can be extended to the whole ball. In addition, it is proved that if complex manifolds $S_1$ and $S_2$ have the property that every locally biholomorphic map of the domain $D$ over $\mathbf C^n$ into $S_j$ can be holomorphically extended to the envelope of holomorphy $\widetilde D$ of $D$, then the product $S_1\times S_2$ possesses the same property.
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