${\mathtt{w}}''(u) + {\mathtt{w}}''(v) \notin \{{\mathtt{lb}}', {\mathtt{ub}}'\}$ for all $(u,v) \in V^{2}$. We also showed that the number of unique-weight vertices under ${\mathtt{w}}''$ is larger than the one under ${\mathtt{w}}'$. This contradicts the assumption on how ${\mathtt{w}}'$ is chosen, and thus we can conclude that no two vertices have the same weight in ${\mathtt{w}}'$.

We continue the proof of Lemma \[lem:nice-representation\]. Let ${\mathtt{w}}'$ be the vertex-weight function in Claim \[clm:nicer-rep2\]. Given ${\mathtt{lb}}^{*}, {\mathtt{ub}}^{*} \in {\mathbb{R}}$ with ${\mathtt{lb}}^{*} < {\mathtt{ub}}^{*}$, let $\rho = ({\mathtt{ub}}^{*} - {\mathtt{lb}}^{*})/({\mathtt{ub}}' - {\mathtt{lb}}')$. For each $v \in V$, we set ${\mathtt{w}}^{*}(v) = \rho \cdot {\mathtt{w}}'(v) - \left(\rho \cdot {\mathtt{lb}}' - {\mathtt{lb}}^{*}\right)/2$. Then, ${\mathtt{w}}^{*}(u) \ne {\mathtt{w}}^{*}(v)$ if $u \ne v$. Now for each $(u,v) \in V^{2}$, $$\begin{aligned}
  & {\mathtt{lb}}^{*} < {\mathtt{w}}^{*}(u) + {\mathtt{w}}^{*}(v) < {\mathtt{ub}}^{*}
  \\
  &\iff {\mathtt{lb}}^{*}
  <
  \rho \cdot ({\mathtt{w}}'(u) + {\mathtt{w}}'(v))
  - \left(\rho \cdot {\mathtt{lb}}' - {\mathtt{lb}}^{*}\right)
  <
  {\mathtt{ub}}^{*}
  \\
  &\iff \rho \cdot {\mathtt{lb}}'
  <
  \rho \cdot ({\mathtt{w}}'(u) + {\mathtt{w}}'(v))
  <
  \rho \cdot {\mathtt{lb}}' + {\mathtt{ub}}^{*} - {\mathtt{lb}}^{*}
  = \rho \cdot {\mathtt{lb}}'  + \rho \cdot ({\mathtt{ub}}' - {\mathtt{lb}}')
  \\
  &\iff {\mathtt{lb}}' 
  <
  {\mathtt{w}}'(u) + {\mathtt{w}}'(v)
  <
  {\mathtt{ub}}'.\end{aligned}$$ Similarly, ${\mathtt{w}}^{*}(u) + {\mathtt{w}}^{*}(v) < {\mathtt{lb}}^{*}$ if and only if ${\mathtt{w}}'(u) + {\mathtt{w}}'(v) < {\mathtt{lb}}'$, and ${\mathtt{w}}^{*}(u) + 