Corollary

Let $(X_\iota)_{\iota \in I}$ and $(Y_\varkappa)_{\varkappa \in K}$ be two families of sets with non-empty index sets I, K. Then

Let $\alpha$, $\beta$ be two distinct objects; apply the formulae of Proposition 8 (with $L = \{\alpha,\ \beta\}$, $J_\alpha = I$, $J_\beta = K$) to the family $((Z_{\lambda,\, \mu})_{\mu \in J_\lambda})_{\lambda \in L}$, where $Z_{\alpha,\, \iota} = X_\iota$ for all $\iota \in I$ and $Z_{\beta,\, \varkappa} = Y_\varkappa$ for each $\varkappa \in K$. By the existence of the canonical bijection of $\coprod_{\lambda \in L} J_\lambda$ onto $I \times K$ (no. 3) and by Proposition 1 of § 4 we obtain the formulae stated.