Proposition 8

Let $(\mathrm{X}_\iota)_{\iota \in \mathrm{I}}$ be a family of mutually disjoint sets, and let $(f_\iota)_{\iota \in \mathrm{I}}$ be a family of functions with the same target F such that the domain of $f_\iota$ is $\mathrm{X}_\iota$ for each $\iota \in \mathrm{I}$. Then there exists exactly one function f with domain $\bigcup_{\iota \in \mathrm{I}} \mathrm{X}_\iota$ and target F which extends each of the functions $f_\iota$ $(\iota \in \mathrm{I})$.

This is a corollary of Proposition 7, since $f_\iota$ and $f_\varkappa$ clearly agree on $\mathrm{X}_\iota \cap \mathrm{X}_\varkappa = \emptyset$ whenever $\iota \neq \varkappa$.