By Eq. 2.4.3, for any $h=0,\ldots,q-1$, the sections $s^{\sigma_{h+1}}$ are constructed from $s^{\sigma_{h}}$ by multiplication by the matrix for the change of basis from $((v_{i})_{i\in L_{\sigma_{h}}},(v_{j})_{j\in J})$ to $((v_{i})_{i\in L_{\sigma_{h+1}}},(v_{j})_{j\in J})$. Thus, by composition, the sections $s^{\sigma}$ only depends on $s^{\sigma_{0}}$ and the change of basis from $((v_{i})_{i\in L_{\sigma_{0}}},(v_{j})_{j\in J})$ to $((v_{i})_{i\in L_{\sigma}},(v_{j})_{j\in J})$. ∎