Corollary 1

Let $(\mathfrak{a}_\iota)_{\iota \in I}$ be a family of cardinals. For each subset J of I we have $\sum_{\iota \in J} \mathfrak{a}_\iota \leqslant \sum_{\iota \in I} \mathfrak{a}_\iota$. If also $\mathfrak{a}_\iota \neq 0$ for all $\iota \in I - J$, then $\mathbf{P}_{\iota \in J} \, \mathfrak{a}_\iota \leqslant \mathbf{P}_{\iota \in I} \, \mathfrak{a}_\iota$.

Put $\mathfrak{b}_\iota = \mathfrak{a}_\iota$ if $\iota \in J$, and $\mathfrak{b}_\iota = 0$ (resp. $\mathfrak{b}_\iota = 1$) if $\iota \in I - J$. Then apply Proposition 14, observing that the relation $\mathfrak{a} \neq 0$ implies $\mathfrak{a} \geqslant 1$.