\begin{enumerate}[label=(\alph*)]
\item $I_i = g_i(V-E_i)$\\
$\Rightarrow \sum_{i=1}^{n}I_i = \sum_{i=1}^{n}g_i(V-E_i) = \sum_{i=1}^{n}(g_i\cdot V) - \sum_{i=1}^{n}(g_i\cdot E_i)\\
= V\sum_{i_1}^{n}g_i - \sum_{i=1}^{n}(g_i\cdot E_i)\\
V\sum_{i_1}^{n}g_i - \sum_{i=1}^{n}(g_i\cdot E_i)\\
\Leftrightarrow V = \frac{\sum_{i=1}^{n}(g_i\cdot E_i)}{\sum_{i=1}^{n}g_i}$
\item Let $E_1 < E_2 (\dagger)$\\
$E_1 = \frac{E_1(g_1+g_2)}{g_1+g_2}=\frac{E_1g_1 + E_1g_2}{g_1+g_2}\overset{\dagger}{\leq}\frac{E_1g_1 + E_2g_1}{g_1+g_2} = E$\\
$E = \frac{E_1g_1 + E_2g_1}{g_1+g_2} \overset{\dagger}{\leq} \frac{E_2g_1 + E_2g_2}{g_1+g_2} = E_2$\qed
\end{enumerate}