Lemma 19.7.23EC9CD63CAB3A

Proof. Take sequences $S^ j,S'^ j$ in $\mathcal{S}(C)$ such that $S^ j\to S$, $S'^ j\to S'$. By 19.7.19, after replacing $S^ j$ by $S^ j\cap S'^ j$, we may assume that $S^ j\subseteq S'^ j$ for each $j$. Then our assertion follows easily. $\square$