= 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.
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