Proposition 9

Let $((X_{\lambda,\, \iota})_{\iota \in J_\lambda})_{\lambda \in L}$ be a family (with index set L) of families of sets. Let $I = \prod_{\lambda \in L} J_\lambda$. Then

and (if $L \neq \emptyset$ and $J_\lambda \neq \emptyset$ for each $\lambda \in L$)

("distributivity" of product over union and over intersection). The first formula is trivially true if $L = \emptyset$ or if $J_\lambda = \emptyset$ for some $\lambda \in L$. If not, let $g$ be an element of $\prod_{\lambda \in L} \left( \bigcap_{\iota \in J_\lambda} X_{\lambda,\, \iota} \right)$. For each $\lambda \in L$ there exists an index $\iota \in J_\lambda$ such that $g(\lambda) \in X_{\lambda,\, \iota}$; in other words, the set $H_\lambda$ of indices $\iota \in J_\lambda$ such that $g(\lambda) \in X_{\lambda,\, \iota}$ is not empty. By Corollary 2 to Proposition 5 there is therefore a functional graph $f$ with domain L such that

for each $\lambda \in L$, i.e., $g(\lambda) \in X_{\lambda,\, f(\lambda)}$. Hence we have $g \in \prod_{\lambda \in L} X_{\lambda,\, f(\lambda)}$ and consequently $g \in \bigcup_{f \in I} \left( \prod_{\lambda \in L} X_{\lambda,\, f(\lambda)} \right)$. Conversely, if

there is a functional graph $f \in I$ such that for every $\lambda \in L$ we have

and, a fortiori, $g(\lambda) \in \bigcup_{\iota \in J_\lambda} X_{\lambda, \iota}$. This completes the proof of the first formula. The proof of the second formula is analogous but simpler, and we leave it to the reader.