Abstract: Consider two domains $D$ and $G$ in $\mathbb{C}^n$, each of which is the product of smoothly bounded domains, and assume that each factor of $D$ satisfies condition R, i.e, the Bergman projection preserves the class of functions smooth up to the boundary. We show that any proper holomorphic map from $D$ to $G$ extends smoothly to the closures, and splits as a product of equidimensional mappings of the factors. We also consider some possible generalization to a class of piecewise smooth domains. This is joint work with Kaushal Verma.