  We consider proper holomorphic mappings of equidimensional pseudoconvex
domains in complex Euclidean space, where both source and target can be
represented as Cartesian products of smoothly bounded domains. It is shown that
such mappings extend smoothly up to the closures of the domains, provided each
factor of the source satisfies Condition R. It also shown that the number of
smoothly bounded factors in the source and target must be the same, and the
proper holomorphic map splits as product of proper mappings between the factor
domains.
