Proposition 6

Let $(X_\iota)_{\iota\in I}$ be a family of sets such that $X_\iota\ne\varnothing$ for all $\iota\in I$. If $g$ is a mapping of $J\subset I$ into $A=\displaystyle\bigcup_{\iota\in I}X_\iota$, such that $g(\iota)\in X_\iota$ for all $\iota\in J$, then there exists an extension $f$ of $g$ to $I$ such that $f(\iota)\in X_\iota$ for all $\iota\in I$.

For each $\iota\in I-J$ let $T_\iota$ denote the term $\tau_\iota(y\in X_\iota)$. Since $X_\iota\ne\varnothing$ by hypothesis, we have $T_\iota\in X_\iota$ for all $\iota\in I-J$ (Chapter I, §4, no. 1). If $G$ is the graph of $g$, the graph $G\cup\left(\displaystyle\bigcup_{\iota\in I-J}\{(\iota,T_\iota)\}\right)$ is the graph of a function which has the required properties, as is immediately verified.