PomeProposition 17.15.35.A44C0F3CE359 Let $\Gamma \in \mathrm{TC}(X,\theta )_{>0}$, $\Gamma '\in \mathrm{TC}(X,\theta ')_{>0}$, $C\in \mathbb {R}$ and $\lambda ,\lambda '>0$, we have
\[ \begin{aligned} \lambda (\Gamma +\Gamma ')=& \lambda \Gamma +\lambda \Gamma ',\\ (\lambda \lambda ')\Gamma =& \lambda (\lambda '\Gamma ),\\ \lambda (\Gamma +C)=& \lambda \Gamma +\lambda C. \end{aligned} \]
Suppose that $(\Gamma ^ i)_{i\in I}$ is a non-empty family in $\mathrm{TC}(X,\theta )_{>0}$ satisfying (17.15.21). Then
\[ \lambda \left( \operatorname *{sup}_{i\in I}{}\! {}^{*}\Gamma ^ i \right)=\operatorname *{sup}_{i\in I}{}\! {}^{*}(\lambda \Gamma ^ i). \]
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