mR\
& R m ( 1+ L\_[k-1]{}\^[-1/8]{} ) ( 1 - 3L\_[k-1]{}\^[-1/6]{} ) R m (1 + L\_[k-1]{}\^[-1/8]{} )\
&&gt; (m, L\_k) |x - y| + | \_[L\_k]{}(u)|, as required for the $(E,m)$-NS property of the ball $\ball_{L_k}(u)$.

If $r'=0$ (resp., $r''=0$), the required bound follows from the subharmonicity of the function $f(x',x'')$ in $x''$ (resp., in $x'$).

\(B) Assume otherwise. Then, owing to assertion (A), the $E$-CNR ball $\ball_{L_k}(u)$ must contain a pair of disjoint $(E,m)$-S balls $\ball_{L_{k-1}}(x)$, $\ball_{L_{k-1}}(y)$. Neither of them is $E$-R, since $\ball_{L_k}(u)$ is $E$-CNR. By virtue of Lemma \[lem:loc.and.NR.imply.NS.IID\], both $\ball_{L_{k-1}}(x)$ and $\ball_{L_{k-1}}(y)$ must be , so that $\ball_{L_k}(u)$ must be $\mathtun$, which contradicts the hypothesis.

\[lem:NT.implies.nloc.IID\] There exists $\tilde L^{(2)}\in\DN$ such that, for all $L_0\ge \tilde L^{(2)}$, if for any $E\in\DR$ a ball $\ball_{L_k}(u)$ contains no pair of disjoint $(E,m)$-S balls of radius $L_{k-1}$, then it is $\mathloc$.

One can proceed as in the proof of the previous lemma, but with the functions $
 f_j(x', x'') = |\psi_j(x') \psi_j(x'')|,
$ where $\psi_j, j\in[1, |\ball_{L_k}(u)|]$, are normalized eigenfunctions of operator $H_{\ball_{L_k}(u)}$. Notice that the $E_j$-non-resonance condition for $\ball_{L_k}(u)$ is not required here, since $\|\psi_j\|=1$, so the function $|f_j|$ is globally bounded by $1$. Let $x,y\in 