Lexicographic embedding of a sumset

Represent each element of $S_1+\cdots +S_n$ by the lexicographically least tuple with that sum. Every coordinate projection of the resulting tuple set is itself lexicographically least for its partial sum: replacing it by a smaller representative would improve the original tuple without changing its sum. Consequently projected cardinalities are at most the corresponding partial sumset cardinalities. Projection inequalities, including the box theorem, therefore give sumset inequalities.