following bound holds true: \[eq:lem.induct.IID\] u\^d L\_[k’]{}\^[-p(1+b)\^[k’]{}]{}. Then, for $L_0>0$ large enough, $\forall\, u\in\DZ^d$ \[eq:lem.induct.next.Tun.IID\] L\_[k+1]{}\^[-p(1+b)\^[k+1]{}]{} and for any pair of disjoint balls $\ball_{L_{k}}(u)$, $\ball_{L_{k}}(v)\subset\ball_{L_{k+1}}(w)$ \[eq:lem.induct.next.S.IID\] L\_k\^[-p(1+b)\^[k+1]{}]{}.

Consider the following events: $$\bal
\CN_{k+1} &:= \{ \text{$\ball_{L_{k+1}}(w)$ is } \mathnloc \} \\
\CS_{k}^{(2)} &:= \{\exists\,E\;\exists\,  \text{ disjoint $(E,m)$-S balls } \ball_{L_{k}}(u),  \ball_{L_{k}}(v)\subset \ball_{L_{k+1}}(w) \} \\
\CR_{k}^{(2)} &:= \{\exists\,E\;\exists\,  \text{ disjoint $E$-PR balls } \ball_{L_{k}}(u),  \ball_{L_{k}}(v)\subset \ball_{L_{k+1}}(w)  \}.
\eal$$ By Lemma \[lem:NT.implies.nloc.IID\], $$\CN_{k+1} \subset \CR_{k}^{(2)} \cup \CS_{k}^{(2)} \equiv
  \CR_{k}^{(2)} \cup \left(\CS_{k}^{(2)}\setminus \CR_{k}^{(2)} \right),$$ and by Wegner-type bound , $\pr{\CR_k^{(2}}\le e^{-L_k^{\beta'}}$, so that it remains to bound $\pr{\CS_{k}^{(2)}\setminus \CR_{k}^{(2)} }$. Fix points $u,v$ with $d(u, v)>2L_k$ and introduce the event \[eq:S2.u.v\] \_[k]{}\^[(2)]{}(u,v) = {E: }. Within the event $\CS_{k}^{(2)}(u,v)\setminus \CR_{k}^{(2)}$, either $\ball_{L_k}(u)$ or $\ball_{L_k}(v)$ must be $E$-CNR; without loss of generality, suppose $\ball_{L_k}(u)$ is $E$-CNR. At the same time, it is $(E,m)$-S, so by assertion (B) of Lemma \[lem:few.nloc.and.few.R.imply.nloc.IID\], $\ball_{L_k}(u)$ must be $\mathtun$, i.e., it must contain a pair of disjoint balls $\ball_{L_{k-1}}(y')$, $\ball_{L_{k-1}}(y'')$. Using the inductive assumption and independence[^2] of random operators $H_{\ball_{L_{k-1}}(y')}$, $H_{\ball_{L_{k-1}}(y'')}$, we can write $$\pr{ \text{$\ball_{L_{k-1}}(y')$ and $\ball_{L_{k-1}}(y')$ are \nloc} }
\le L_{k-1}^{-2p(1+b)^{k-1}} \le  L_{k+1}^{- \frac{2p}{\alpha^2} (1+b)^{k-1} }.$$ The number of all pairs $y', y''$ is bounded by $|\ball_{L_{k+1}}(u)|^2/2$, 