element with $t''_{k+1}=(v'_{1}, \dots , v'_{m_{k}+1})$ such that $v'_{i} =v$ and $v'_{l} =0$ for all $l\neq i$. By Lemma \[ideal\], we have that $$\begin{aligned}
 \lefteqn{(\lambda_{((t_1,\dots,t_n),(s_1, \dots , s_n))}-{\operatorname{id}})(c'_{k+1})}\\
 &=&((t_1,\dots,t_n),(s_1, \dots , s_n))((t''_1,\dots ,t''_n),(0,\dots,0))\\
 &&-((t_1,\dots,t_n),(s_1, \dots , s_n))-((t''_1,\dots
 ,t''_n),(0,\dots,0))\\
 &=&((t_1+f_{s_n}^{(n,1)}(t''_1),\dots,t_n+f_{s_{n-1}}^{(n-1,n)}(t''_n)),(s_1, \dots , s_n))\\
 &&-((t_1+t''_1,\dots,t_n+t''_n),(s_1+b'_1(t_1,t''_1), \dots , s_n+b'_n(t_n,t''_n)))\\
 &=&((f_{s_n}^{(n,1)}(t''_1)-t''_1,\dots,f_{s_{n-1}}^{(n-1,n)}(t''_n)-t''_n),\\
 &&\quad (-b'_1(t_1,t''_1)-b'_1(f_{s_n}^{(n,1)}(t''_1)-t''_1,t_1+t''_1),\\
&&\quad  \dots ,
-b'_n(t_n,t''_n)-b'_n(f_{s_{n-1}}^{(n-1,n)}(t''_n)-t''_n,t_n+t''_n)))\in
 I\cap A_{k+1}
 \end{aligned}$$ and $$f_{s_{k}}^{(k,k+1)} (t''_{k+1}) -t''_{k+1} =
  (0,\dots , 0, f_{k+1}^{\mu_{i}} (v) -v, 0, \dots , 0).$$ Hence Claim 3 follows.

[*Claim 4*]{}: If $\mu_{r_{k}}\neq 0$ then $c_{k+1}\in I$ for some $c_{k+1}\in A_{k+1}$ such that $c_{k+1}=((t'_1,\dots, t'_n),(s'_1,\dots,s'_n))$, $t'_{k+1}=(v_{1}, \dots , v_{m_{k+1}})$, and some $v_{l}\neq 0$.\
As in the proof of Claim 3, there exists $v\in V_{k+1}$ such that $f_{k+1}^{\mu_{r_k}}(v)\neq v$. Let $$c'_{k+1} =((t''_1,\dots ,t''_n),(0,\dots, 0))\in A_{k+1}$$ be the element with $t''_{k+1}=(v'_{1}, \dots , v'_{m_{k}+1})$ such that $v'_{r_k} =v$ and $v'_{l} =0$ for all $l\neq r_k$. As in the proof of Claim 3, we have that $$\begin{aligned}
 \lefteqn{(\lambda_{((t_1,\dots,t_n),(s_1, \dots , s_n))}-{\operatorname{id}})(c'_{k+1})}\\
 &=&((f_{s_n}^{(n,1)}(t''_1)-t''_1,\dots,f_{s_{n-1}}^{(n-1,n)}(t''_n)-t''_n),\\
 &&\quad (-b'_1(t_1,t''_1)-b'_1(f_{s_n}^{(n,1)}(t''_1)-t''_1,t_1+t''_1),\\
&&\quad  \dots ,
-b'_n(t_n,t''_n)-b'_n(f_{s_{n-1}}^{(n-1,n)}(t''_n)-t''_n,t_n+t''_n)))\in
 I\cap A_{k+1}
 \end{aligned}$$ and $$f_{s_{k}}^{(k,k+1)} (t''_{k+1}) -t''_{k+1} =
  (0,\dots , 0, f_{k+1}^{\mu_{r_k}} (v) -v, 0, \dots , 0).$$ Hence Claim 4 follows.

We are now in a position to prove the sufficiency of the conditions. By Claims 1, 2, 