+By switching the indices of \eqref{subsub:hk02} and adding the resulting equation to \eqref{subsub:hk02}, the contradiction
+\begin{equation}
+E_1 + E_2 < E_2 + E_1 +
+\underbrace{
+\int n(\vec{r}) \left( V_1(\vec{r})-V_2(\vec{r}) \right) d\vec{r} +
+\int n(\vec{r}) \left( V_2(\vec{r})-V_1(\vec{r}) \right) d\vec{r}
+}_{=0}
+\end{equation}
+is revealed.