We show the embedding provided by the Hahn Embedding Theorem of an ordered Abelian group into a lexicographic functions space is coinitiality preserving. From this we strengthen Hajek's Completeness theorem for predicate product logic. We conclude that the set of tautologies for a lexicographic function space over a set S which is not initially scattered is recursively enumerable. By contrast we conclude that the set of tautologies for a lexicographic function space over a set S which is initially scattered is not arithmetical.