Proposition: Product of Two Sums (Generalized Distributivity Rule)
Given two sums $\sum_{i=1}^n x_i$ and $\sum_{j=1}^m y_j$ of the elements $x_i,y_j\in R$ of a unit ring $(R,+\cdot)$, their product is given by
$$\left(\sum_{i=1}^n x_i\right)\cdot \left(\sum_{j=1}^m y_i\right)=\sum_{i=1}^n\sum_{j=1}^m x_iy_j.$$
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References
Bibliography
 Forster Otto: "Analysis 1, Differential und Integralrechnung einer Veränderlichen", Vieweg Studium, 1983