# Change of Index Variable of Summation

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## Theorem

- $\displaystyle \sum_{R \left({i}\right)} a_i = \sum_{R \left({j}\right)} a_j$

where $\displaystyle \sum_{R \left({i}\right)} a_i$ denotes the summation over $a_i$ for all $i$ that satisfy the propositional function $R \left({i}\right)$.

## Proof

## Sources

- 1997: Donald E. Knuth:
*The Art of Computer Programming: Volume 1: Fundamental Algorithms*(3rd ed.) ... (previous) ... (next): $\S 1.2.3$: Sums and Products: $(5)$