A generalisation of Läuchli's lemma

Läuchli showed in the absence of the Axiom of Choice () that and, consequently, for all infinite cardinals , where and are the cardinalities of the set of finite subsets and the power set, respectively, of a set which is of cardinality . In this article, we give a generalisation of a simple form of Läuchli's lemma from which several results can be obtained. That is, in the latter equation can be replaced by other cardinals which are equal to in but not in , for example, and , the cardinalities of the set of permutations and the set of partitions, respectively, of a set which is of cardinality .