Exchange of Order of Indexed Summations/Rectangular Domain

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 $b$. Let $D = \left[{a \,.\,.\, b}\right] \times \left[{c \,.\,.\, d}\right]$ be the cartesian product.

Let $f : D \to \mathbb A$ be a mapping

Then we have an equality of indexed summations:
 * $\displaystyle \sum_{i \mathop = a}^b \sum_{j \mathop = c}^d f(i, j) = \sum_{j \mathop = c}^d \sum_{i \mathop = a}^b f(i, j)$

Outline of Proof
We use induction on $d$. In the induction step, we use Indexed Summation of Sum of Mappings.

Proof
The proof goes by induction on $d$.

Basis for the Induction
Let $d < c$.

Then the indexed summation in the is zero.

By Indexed Summation of Zero, so is the.

This is our basis for the induction.

Induction Step
Let $d \geq c$.

We have:

By the Principle of Mathematical Induction, the proof is complete.

Also see

 * Exchange of Order of Summations over Finite Sets