Corollary 3

Let $\mathfrak{a}$ be an infinite cardinal and let $(\mathfrak{a}_\iota)_{\iota \in \mathrm{I}}$ be a family of cardinals $\leqslant \mathfrak{a}$ whose index set I has a cardinal $\leqslant \mathfrak{a}$. Then $\sum\limits_{\iota \in \mathrm{I}} \mathfrak{a}_\iota \leqslant \mathfrak{a}$; and if $\mathfrak{a}_\iota = \mathfrak{a}$ for at least one index $\iota \in \mathrm{I}$, then $\sum\limits_{\iota \in \mathrm{I}} \mathfrak{a}_\iota = \mathfrak{a}$.

Let $\mathfrak{b}$ be the cardinal of I; then we have $\sum\limits_{\iota \in \mathrm{I}} \mathfrak{a}_\iota \leqslant \mathfrak{a}\mathfrak{b} \leqslant \mathfrak{a}^2 = \mathfrak{a}$ (§ 3, no. 6, Proposition 14), and $\sum\limits_{\iota \in \mathrm{I}} \mathfrak{a}_\iota \geqslant \mathfrak{a}_\varkappa$ for all $\varkappa \in \mathrm{I}$.