o [(\[eq:IR-xbdNN\])]{} and the diagrammatic bound [(\[eq:piNbd\])]{}. It thus remains to show [(\[eq:pi-kbd\])]{} for $\pi_\Lambda^{{\scriptscriptstyle}(j)}(x)$ with $x\ne o$ and $j\ge1$.

The idea of the proof is somewhat similar to that of Proposition \[prp:exp-bootstrap\](iii) explained above. First, we take $|a_m|\equiv\max_n|a_n|$ from the lowermost path and $|a'_l|\equiv\max_n|a'_n|$ from the uppermost path of a bounding diagram. Note that, by [(\[eq:|x|-max\])]{}, $|a_m|$ and $|a'_l|$ are both bigger than $\frac1{j+1}|x|$. That is, $|a_m|^{-q}$ and $|a'_l|^{-q}$ are both bounded from above by $(j+1)^q|x|^{-q}$. If the path corresponding to $a_m$ in the $m^\text{th}$ block consists of $N$ segments, we take the “longest” segment whose end-to-end distance is therefore bigger than $\frac1{N(j+1)}|x|$. That is, the corresponding two-point function is bounded by $\lambda_0
N^q(j+1)^q|x|^{-q}$. Here, $N$ depends on the parity of $m$, as well as on $i\ge0$ for $P_{\Lambda;u,v}^{\prime\prime{{\scriptscriptstyle}(i)}}$ (or $P_{\Lambda;u}^{\prime{{\scriptscriptstyle}(i)}}$ if $m=0$ or $j$) and the location of $u,v$ in each diagram, and is at most $N\leq O(i+1)$. However, the number of nonzero chains of bubbles contained in each diagram of $P_{\Lambda;u}^{\prime{{\scriptscriptstyle}(i)}}$ and $P_{\Lambda;u,v}^{\prime\prime{{\scriptscriptstyle}(i)}}$ is $O(i)$, and hence their contribution would be $O(\theta_0)^{O(i)}$. This compensates the growing factor of $N^q$, and therefore we will not have to take the effect of $N$ seriously. The same is true for $a'_l$, and we refrain from repeating the same argument.

Next, we take the “longest” segment, denoted $a''$, among those which together with $a'_l$ (or a part of it) form a “loop”; a similar observation was used to obtain [(\[eq:IRSchwarz-bdii\])]{}. The loop consists of segments contained in the $l^\text{th}$ block and possibly in the $(l-1)^\text{st}$ block, and hence the number of choices for $a''$ is at most $O(i_{l-1}+i_l+1)$, where $i_l$ is the index