If $s$ and $s^{\prime}$ are two toric sections, then there is an element $m\in M$ such that $s^{\prime}=\chi^{m}s$. Since for any element $t\in\mathbb{S}^{{\text{\rm an}}}$ we have $|\chi^{m}(t)|=1$, if the function $\|s(p)\|$ is $\mathbb{S}^{{\text{\rm an}}}$-invariant, then the function $\|s^{\prime}(p)\|=\|\chi^{m}(p)s(p)\|$ is also $\mathbb{S}^{{\text{\rm an}}}$-invariant. ∎