Proposition 6

Let $(\mathfrak{a}_\iota)_{\iota \in \mathrm{I}}$ be a family of cardinals, and let $\mathrm{J}$ (resp. $\mathrm{K}$) be a subset of $\mathrm{I}$ such that $\mathfrak{a}_\iota = 0$ for all $\iota \notin \mathrm{J}$ (resp. $\mathfrak{a}_\iota = 1$ for all $\iota \notin \mathrm{K}$). Then

The proposition is obvious as regards the sum, for the sum $\mathrm{S}_\mathrm{I}$ of the family of sets $(\mathfrak{a}_\iota)_{\iota \in \mathrm{I}}$ is equipotent to the union of the sum $\mathrm{S}_\mathrm{J}$ of the family $(\mathfrak{a}_\iota)_{\iota \in \mathrm{I}}$ and the empty set, and hence equipotent to $\mathrm{S}_\mathrm{J}$. The assertion concerning products follows from the fact that the projection $\mathrm{pr}_\mathrm{K}$ of the product set $\prod_{\iota \in \mathrm{I}} \mathfrak{a}_\iota$ onto the partial product $\prod_{\iota \in \mathrm{K}} \mathfrak{a}_\iota$ is a bijection (Chapter II, § 5, no. 5, Remark 1).