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We generalize to the relations $(\lambda, \mu) \stackrel{\kappa}{\Rightarrow}(\lambda', \mu')$ and $\alm (\lambda, \mu) \stackrel{\kappa}{\Rightarrow} \alm(\lambda', \mu')$ some results obtained in Parts II and IV. We also present amulti-cardinal version.

Combinatorial and model-theoretical principles related to regularity of ultrafilters and compactness of topological spaces. V