Proposition 10

Let $(\mathrm{X}_\iota)_{\iota \in \mathrm{I}}$ be a family of mutually disjoint sets. Let A be its union and S its sum. Then there exists a bijective mapping of A onto S.

For each $\iota \in \mathrm{I}$, let $f_\iota$ be a bijection of $\mathrm{X}_\iota$ onto $\mathrm{X}_\iota \times \{\iota\}$. By virtue of Proposition 8, there exists a mapping $f$ of A into S which extends all the mappings $f_\iota$. It is immediately verified that $f$ is a bijective mapping of A onto S.

By abuse of language, a set E is said to be the sum of a family of sets $(\mathrm{X}_\iota)_{\iota \in \mathrm{I}}$ if there exists a bijection of E onto the sum of the family, as defined in Definition 8.

Note that if $(\mathrm{X}_\iota)_{\iota \in \mathrm{I}}$ is an arbitrary family of sets, the argument of Proposition 10 shows that there exists a mapping of the sum S onto the union A.