Proposition 10

Let $(X_{\iota, \varkappa})_{(\iota, \varkappa) \in I \times K}$ be a family of sets whose index set is the product of two sets I and K. If $K \neq \emptyset$, we have

Both sides of the equality to be proved are functional graphs. A graph $f$ belongs to the left-hand side if and only if, for each $\varkappa \in K$, $f \in \prod_{\iota \in I} X_{\iota, \varkappa}$; that is, if and only if $f(\iota) \in X_{\iota, \varkappa}$ for all $(\iota, \varkappa) \in I \times K$. For $f$ to belong to the right-hand side it is necessary and sufficient that $f(\iota) \in \bigcap_{\varkappa \in K} X_{\iota, \varkappa}$ for each $\iota \in I$, i.e., that $f(\iota) \in X_{\iota, \varkappa}$ for each pair $(\iota, \varkappa) \in I \times K$. This completes the proof.