Exchange of Order of Summation with Dependency on Both Indices/Example

Theorem
Let $n \in \Z$ be an integer.

Let $R: \Z \to \left\{ {\mathrm T, \mathrm F}\right\}$ be the propositional function on the set of integers defining:
 * $\forall i \in \Z: R \left({i}\right) := \left({n = k i \text { for some } k \in \Z}\right)$

Let $S: \Z \times \Z \to \left\{ {\mathrm T, \mathrm F}\right\}$ be a propositional function on the Cartesian product of the set of integers with itself defining:
 * $\forall i, j \in \Z: S \left({i, j}\right) := \left({1 \le j < i}\right)$

Consider the summation:


 * $\displaystyle \sum_{R \left({i}\right)} \sum_{S \left({i, j}\right)} a_{i j}$

Then:


 * $\displaystyle \sum_{R \left({i}\right)} \sum_{S \left({i, j}\right)} a_{i j} = \sum_{S' \left({j}\right)} \sum_{R' \left({i, j}\right)} a_{i j}$

where:
 * $S' \left({j}\right)$ denotes the propositional function:
 * $\forall j \in \Z: S' \left({j}\right) := \left({1 < j \le n}\right)$


 * $R' \left({i, j}\right)$ denotes the propositional function:
 * $\forall i, j \in \Z: R' \left({i, j}\right) := \left({n = k i \text { for some } k \in \Z \text { and } i > j}\right)$

Proof
From Exchange of Order of Summation with Dependency on Both Indices:


 * $S' \left({j}\right)$ denotes the propositional function:
 * there exists an $i$ such that both $R \left({i}\right)$ and $S \left({i, j}\right)$ hold
 * $R' \left({i, j}\right)$ denotes the propositional function:
 * both $R \left({i}\right)$ and $S \left({i, j}\right)$ hold.

The definition of $R' \left({i, j}\right)$ follows immediately:


 * $R \left({i}\right) := \left({n = k i \text { for some } k \in \Z}\right)$

and:
 * $S \left({i, j}\right) := \left({1 \le j < i}\right)$

Then:

By Integer Absolute Value not less than Divisors, it follows from $R \left({i}\right)$ that $i \le n$.

That is, for $S' \left({j}\right)$ to hold, $i \le n$.

But for all $j \in \Z$ such that $1 \le j < n$ it follows that $i = n$ fulfils the condition that $n = k i$.

Hence:
 * $S' \left({j}\right) := \left({1 < j \le n}\right)$