Proposition: Rule of Combining Different Sets of Indices
If \(K\) and \(L\) are any finite sets of integers, \(k\in K\cup L\), and \(a_k\in F\) any elements of a given field \((F, +, \cdot)\), then
\[\sum_{k\in K} a_k + \sum_{k\in L} a_k=\sum_{k\in K\cap L} a_k + \sum_{k\in K\cup L} a_k\]
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References
Bibliography
 Graham L. Ronald, Knuth E. Donald, Patashnik Oren: "Concrete Mathematics", AddisonWesley, 1994, 2nd Edition