Exchange of Order of Indexed Summations
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Theorem
Let $\mathbb A$ be one of the standard number systems $\N, \Z, \Q, \R, \C$.
Let $a, b, c, d \in \Z$ be integers.
Let $\closedint a b$ denote the integer interval between $a$ and $b$.
Rectangular Domain
Let $D = \closedint a b \times \closedint c d$ be the cartesian product.
Let $f: D \to \mathbb A$ be a mapping
Then we have an equality of indexed summations:
- $\ds \sum_{i \mathop = a}^b \sum_{j \mathop = c}^d \map f {i, j} = \sum_{j \mathop = c}^d \sum_{i \mathop = a}^b \map f {i, j}$