and (\[LemmaBoxPropertyNotA\]) are equivalent. We prove the equivalence of conditions (\[LemmaBoxConditionNonAme\]) and (\[LemmaBoxPropertyWES\]). Let $K$ be a symmetric generating set of $G$ containing $1_G$. Define a controlled set $T_K$ by the action of $K$, i.e., $T_K = \{ (g x, x) \ 
|\ g \in K, x \in \square G \}$. The controlled set $T_K^{\circ n}$ corresponds to the action of $K^n$, namely, $$\begin{aligned}
T_K^{\circ n} = \{ (g_1 g_2 \cdots g_n x, x) \ 
|\ g_1, g_2, \cdots, g_n \in K, x \in \square G \}.\end{aligned}$$

Suppose that $G$ is amenable. For every controlled set $T$ of $\square G$, there exists a natural number $n$ such that $T \subseteq T_K^{\circ n}$. For an arbitrary positive number $\epsilon$, there exists an $\epsilon$-Følner set $F \subseteq G$ for the action of $K^n$. We mean by $\epsilon$-Følner that $\sharp(F) < \infty$ and that $\sharp(K^n F) < (1 + \epsilon) \sharp (F)$. If $m$ is large enough, then the quotient map $q_m \colon G \rightarrow G / H_m$ is injective on the subset $K^n F \subseteq G$. For large $m$, we have $$\begin{aligned}
     \sharp(T[q_m(F)]) 
\le \sharp(T_K^{\circ n}[q_m(F)]) 
 =  \sharp(q_m(K^n F)) 
 <  (1 + \epsilon) \sharp (q_m(F)).\end{aligned}$$ This implies the inequality $$\begin{aligned}
\liminf_{m \rightarrow 