\chi}} (\beta _\nu )-1}
  ({\rho ^{\chi}} (\beta _\nu )K_{\beta _\nu }-
  \chi (\beta _\nu ,\beta _\nu )^t L_{\beta _\nu })^{N_{\nu ,t}}$$ for some $N_{\nu ,t}\in {\mathbb{N}}_0$ and an element $f\in {\Bbbk }[K_i,L_i\,|\,i\in I]$ which is invertible on ${\mathbb{T}}$. In particular, $\det ^\chi _{\alpha }\not=0$. We finish the proof of the theorem by showing that $N_{\nu ,t}\ge {P}^\chi ({\alpha },\beta _\nu ;t)$ for all $\nu \in \{1,2,\dots ,n\}$ and $t\in \{1,2,\dots ,{b^{\chi}} (\beta _\nu )-1\}$. The essential ingredients will be Lemmata \[le:subfch\] and \[le:detXfactor\].

Let $\nu \in \{1,2,\dots ,n\}$ and $t\in \{1,2,\dots ,{b^{\chi}} (\beta _\nu )-1\}$. Let $w=1_\chi {\sigma }_{i_1}{\sigma }_{i_2}\cdots {\sigma }_{i_{\nu -1}}$. Then $${{\mathcal{U}}^0}={\Bbbk }[K_{w({\alpha }_j)},K_{w({\alpha }_j)}^{-1},
  L_{w({\alpha }_j)},L_{w({\alpha }_j)}^{-1}\,|\,j\in I]$$ and $w({\alpha }_{i_\nu })=\beta _\nu $. Let $$B ={\Bbbk }[L_{\beta _\nu },L_{\beta _\nu }^{-1},
  K_{w({\alpha }_j)},K_{w({\alpha }_j)}^{-1},
  L_{w({\alpha }_j)},L_{w({\alpha }_j)}^{-1}\,|\,j\in I \setminus \{i_\nu \}]$$ and $x={\rho ^{\chi}} (\beta _\nu )K_{\beta _\nu }-
  \chi (\beta _\nu ,\beta _\nu )^t L_{\beta _\nu }$. Then $${{\mathcal{U}}^0}\simeq B[x,(x+
  \chi (\beta _\nu ,\beta _\nu )^t L_{\beta _\nu })^{-1}].$$ Let $X'=(x'_{ij})_{i,j\in \{1,2,\dots ,k\}}\in ({{\mathcal{U}}^0})^{k\times k}$ with $$x'_{ij}={\mathrm{Sh}}(F'_i,F'_j) \quad \text{for all $i,j\in \{1,2,\dots ,k\}$.}$$ Let $l\in {\mathbb{Z}}$ such that $K_{\beta _\nu }^l X'\in B[x]^{k\times k}$, and let $X=K_{\beta _\nu }^l X'$. By Lemma \[le:subfch\] and Eq.  there is a non-empty open subset of the variety of $B\simeq
  {{\mathcal{U}}^0}/(x)$ such that $\mathrm{rk}\,X(0)_p\le k-{P}^\chi ({\alpha },\beta _\nu ;t)$ for all $p$ in this set. By Lemma \[le:detXfactor\], $\det X=x^{{P}^\chi ({\alpha },\beta _\nu ;t)}b'$ for some $b'\in
  B[x]$. In particular, $x^{{P}^\chi ({\alpha },\beta _\nu ;t)}$ is a factor of $\det ^\chi _{\alpha }$, and the proof of the theorem is compl