PomeProposition 22.1.38.938848E420F9 Let $(\Gamma ^ i)_{i\in I}$ be a non-empty family in $\mathrm{PSH}^{\mathrm{NA}}(X,\theta )$ satisfying (22.1.35). Let $C\in \mathbb {R}$. Then
\[ \operatorname *{sup}_{i\in I}{}\! {}^{*} (\Gamma ^ i+C)=\operatorname *{sup}_{i\in I}{}\! {}^{*} \Gamma ^ i+C. \]
Suppose that $(\Gamma '^ i)_{i\in I}$ is another family in $\mathrm{PSH}^{\mathrm{NA}}(X,\theta ')$ satisfying (22.1.35). Suppose that $\Gamma ^ i\leq \Gamma '^ i$ for all $i\in I$. Then
\[ \operatorname *{sup}_{i\in I}{}\! {}^{*}\Gamma ^ i\leq \operatorname *{sup}_{i\in I}{}\! {}^{*}\Gamma '^ i. \]
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