P2760
By direct computation
\begin{equation}
\begin{split}
\left( \sum_{n = 1}^N x_n \right) \left( \sum_{m = 1}^M y_m \right)
& = \left( x_1 + x_2 + \cdots + x_N \right) \left( \sum_{m = 1}^M y_m \right) \\
& = x_1 \sum_{m = 1}^M y_m + x_2 \sum_{m = 1}^M y_m + \cdots + x_N \sum_{m = 1}^M y_m \\
& = \sum_{m = 1}^M x_1 y_m + \sum_{m = 1}^M x_2 y_m + \cdots + \sum_{m = 1}^M x_N y_m \\
& = \sum_{n = 1}^N \sum_{m = 1}^M x_n y_m
\end{split}
\end{equation}
$\square$
