A generalisation of Läuchli's lemma

Läuchli showed in the absence of the Axiom of Choice ($\mathrm{AC}$) that ${\left(2^{fin\left(m\right)}\right)}^{{\aleph }_0}=2^{fin\left(m\right)}$ and, consequently, $2^{2^m}+2^{2^m}=2^{2^m}$ for all infinite cardinals $m$, where $fin\left(m\right)$ and $2^m$ are the cardinalities of the set of finite subsets and the power set, respectively, of a set which is of cardinality $m$. In this article, we give a generalisation of a simple form of Läuchli's lemma from which several results can be obtained. That is, $2^m$ in the latter equation can be replaced by other cardinals which are equal to $2^m$ in $\mathrm{ZF}C$ but not in $\mathrm{ZF}$, for example, $m!$ and $Part\left(m\right)$, the cardinalities of the set of permutations and the set of partitions, respectively, of a set which is of cardinality $m$.