The cones $\sigma$ and $\sigma^{\prime}$ correspond to adjacent vertices in $\Gamma$. Thus, by Lemma 2.5.1 we construct $s^{\sigma}$ from any path joining $\sigma_{0}$ to $\sigma$, and $s^{\sigma^{\prime}}$ from $s^{\sigma}$ by the relation $s^{\sigma^{\prime}}=M_{\mathcal{B^{\prime}}\mathcal{B}}\,s^{\sigma}$ in Eq. 2.4.3.

The functions $\chi^{\varepsilon}$ transform into $\chi^{\varepsilon^{\prime}}$ via the change of dual bases, which is given by $\varepsilon^{\prime}=M_{\mathcal{B^{\prime}}\mathcal{B}}\,\varepsilon$ in Eq. 2.4.1. Comparing the two formulas, it follows that $f_{\sigma}=f_{\sigma^{\prime}}$ on $\mathfrak{X}_{\sigma}\cap\mathfrak{X}_{\sigma^{\prime}}$. ∎