Lemma 16.28.63C172793BD93F

Proof. Fix $b,s\geq 1$ and put $h=Q_{bs}(u,\varphi )$. Its defining inequality gives, off the pole sets,

The left-hand side is $\theta $-plurisubharmonic, so it is at most $Q_ b(u,\varphi )$. Hence $h\in \mathcal A_ s(Q_ b(u,\varphi ),\varphi )$ for every $s$. Also $Q_ b(u,\varphi )\leq u\leq \varphi $, so $Q_ b(u,\varphi )\preceq \varphi $. The inequalities between convex combinations extend across the poles by lemma 5.24.3, which proves the first assertion.

For the second assertion, choose a constant $C$ such that $u-C\leq \varphi $ and $w-C\leq \varphi $. Since $Q_ b(u-C,w-C)=Q_ b(u,w)-C$ and class membership is unchanged by translation, we may assume $u,w\leq \varphi $. Because $u\leq w\leq \varphi $, the obstacle for $Q_ b(u,w)$ lies above that for $Q_ b(u,\varphi )$ and below $u$. Thus

Since $Q_ b(u,\varphi )\in \mathcal E_\varphi $ by the first assertion, the stability under increase proved in proposition 16.28.62 gives the conclusion. $\square$