Lemma 16.8.13005A077A0167

Proof. Take a closed real smooth $(1,1)$-form $\theta $ on $X$ such that $\varphi ,\psi \in \mathrm{PSH}(X,\theta )_{>0}$. We only prove the $P$ case, the $\mathcal{I}$ case is similar.

2 $\implies $ 1. Assume 2. By 2 and example 16.8.6, $P_{\theta }[\varphi \lor \psi ]=P_{\theta }[\psi ]\sim _ P \psi $. But $\varphi \preceq P_{\theta }[\varphi \lor \psi ]$, so 1 follows.

1 $\implies $ 2. Assume 1. We may assume that $\varphi ,\psi $ are both model by 16.8.11. Then $\varphi \leq \psi $ and 2 follows. $\square$