Corollary 16.8.1669E7C4B20FAE

Proof. It suffices to show that for any $\psi \in \mathrm{PSH}(X,\theta )$ with $\psi \leq 0$ and $\psi \preceq _{\mathcal{I}}\varphi $, we have $\psi \leq P_{\theta }[\varphi ]_{\mathcal{I}}$. By 16.8.13 and proposition 16.6.11, we know that $P_{\theta }[\varphi ]_{\mathcal{I}}\lor \psi \sim _{\mathcal{I}}\varphi $. Therefore, $P_{\theta }[\varphi ]_{\mathcal{I}}\lor \psi $ is a candidate in the definition of $P_{\theta }[\varphi ]_{\mathcal{I}}$ (cf. (16.6.8)), so

We conclude the proof. $\square$