{\mathtt{ub}}&< 2 \cdot \min\{ \min\{{\mathtt{w}}_{G}(v) \mid v \in V\}, \min\{{\mathtt{w}}_{H}(w) \mid w \in X \cup Y \}\} + \alpha, \\
  {\mathtt{lb}}&> 2 \cdot \max\{ \max\{{\mathtt{w}}_{G}(v) \mid v \in V\}, \max\{{\mathtt{w}}_{H}(w) \mid w \in X \cup Y \}\} - \alpha.\end{aligned}$$ Now, we set ${\mathtt{w}}\colon (V \cup X \cup Y) \to {\mathbb{R}}$ as follows: $${\mathtt{w}}(w) = 
  \begin{cases}
    {\mathtt{w}}_{G}(w)     & \text{if } w \in V, \\
    {\mathtt{w}}_{H}(w) + \alpha & \text{if } w \in X, \\
    {\mathtt{w}}_{H}(w) - \alpha & \text{if } w \in Y.
  \end{cases}$$

We show that ${\mathtt{w}}$, ${\mathtt{lb}}$, and ${\mathtt{ub}}$ define the disjoint union $(V \cup X \cup Y, E_{G} \cup E_{H})$ of $G$ and $H$. For two vertices $u,v \in V$, we have ${\mathtt{w}}(u) + {\mathtt{w}}(v) = {\mathtt{w}}_{G}(u) + {\mathtt{w}}_{G}(v)$. Similarly, for $x \in X$ and $y \in Y$, we have ${\mathtt{w}}(x) + {\mathtt{w}}(y) = {\mathtt{w}}_{H}(x) + {\mathtt{w}}_{H}(y)$. For $x_{1}, x_{2} \in X$, we have ${\mathtt{w}}(x_{1}) + {\mathtt{w}}(x_{2}) = {\mathtt{w}}_{H}(x_{1}) + {\mathtt{w}}_{H}(x_{2}) + 2\alpha > {\mathtt{ub}}$. For $y_{1}, y_{2} \in Y$, we have ${\mathtt{w}}(y_{1}) + {\mathtt{w}}(y_{2}) = {\mathtt{w}}_{H}(y_{1}) + {\mathtt{w}}_{H}(y_{2}) - 2\alpha < {\mathtt{lb}}$. For $v \in V$ and $x \in X$, we have ${\mathtt{w}}(v) + {\mathtt{w}}(x) = {\mathtt{w}}_{G}(v) + {\mathtt{w}}_{H}(x) + \alpha > {\mathtt{ub}}$. Finally, for $v \in 