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It is known that the satisfiability problems of the product logics K4xS5 andS4xS5 are NEXPTIME-hard and that the satisfiability problem of the logic SSL ofsubset spaces is PSPACE-hard. We improve these lower bounds for the complexityof these problems by showing that all three problems are EXPSPACE-hard underlogspace reduction. In another paper we show that these problems are in ESPACE.This shows that all three problems are EXPSPACE-complete.

EXPSPACE-Completeness of the Logics K4xS5 and S4xS5 and the Logic of Subset Spaces, Part 2: EXPSPACE-Hardness