i<κ λi/I, where κ < mini<κ λi. Here we prove this theorem under weaker assumptions such as wsat(I) < mini<κ λi, where wsat(I) is the minimal θ such that κ cannot be delivered to θ sets / ∈ I (or even slightly weaker condition). We also look at the existence of exact upper bounds relative to <I (<I −eub) as well as cardinalities of reduced products and the cardinals TD(λ). Finally we apply this to the problem of the depth of ultraproducts (and reduced products) of Boolean algebras.