Exchange of Order of Indexed Summations

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 $\left[{a \,.\,.\, b}\right]$ denote the integer interval between $a$ and $b$.

General Domain
Let $f, g : \left[{a \,.\,.\, b}\right] \to \Z$ be mappings.

Let $D = \displaystyle \bigcup_{i \mathop \in \left[{a \,.\,.\, b}\right]} \{i\} \times \left[{f(i) \,.\,.\, g(i)}\right]$ where:
 * $\cup$ denotes union
 * $\times$ denotes cartesian product

Let $h : D \to \mathbb A$ be a mapping.

For $i \in \Z$, let:
 * $C_1(i) \subset D$ be the first cylinder of $D$ above $i$.

Also see

 * Exchange of Order of Summations over Finite Sets