Corollary 5.15.19030D740FC9C6

Proof. It suffices to show that for any $x\in X$ such that $\varphi (x)>\psi (x)$, the same holds on an $\mathcal{F}$-neighborhood $U$ of $x$. Observe that $\varphi (x)\neq -\infty $. If $\psi (x)\neq -\infty $, then it suffices to apply proposition 5.15.18. Otherwise, take

This is an $\mathcal{F}$-open neighborhood of $x$ contained in $\{ \varphi >\psi \} $. $\square$