It is known that the satisfiability problems of the product logics K4xS5 and
S4xS5 are NEXPTIME-hard and that the satisfiability problem of the logic SSL of
subset spaces is PSPACE-hard. We improve these lower bounds for the complexity
of these problems by showing that all three problems are EXPSPACE-hard under
logspace reduction. In another paper we show that these problems are in ESPACE.
This shows that all three problems are EXPSPACE-complete