\begin{align*}
    \mathbf{U} = & \ \mathbf{C} + (\frac{R}{R})(\mathbf{U} - \mathbf{C})\\
    = & \ \mathbf{C} + (\frac{1}{RW} \sum_{i=1}^{n}{w_i r_i})(\mathbf{U} - \mathbf{C}) \\
    = & \ \frac{1}{W} \sum_{i=1}^{n}{w_i \mathbf{c}_i} + \frac{1}{W} \sum_{i=1}^{n}{w_i \frac{r_i}{R}(\mathbf{U} - \mathbf{C})} \\
    = & \ \frac{1}{W} \sum_{i=1}^{n}{w_i (\mathbf{c}_i + \frac{r_i}{R}(\mathbf{U} - \mathbf{C}))} \\
    = & \ \frac{1}{W} \sum_{i=1}^{n}{w_i \mathbf{u}_i}.
\end{align*}
