Sum of Summations over Overlapping Domains/Example

Example of Sum of Summations over Overlapping Domains

 * $\displaystyle \sum_{1 \mathop \le j \mathop \le m} a_j + \sum_{m \mathop \le j \mathop \le n} a_j = \left({\sum_{1 \mathop \le j \mathop \le n} a_j}\right) + a_m$

Proof
Let $R \left({j}\right)$ be the propositional function $1 \mathop \le j \mathop \le m$.

Let $S \left({j}\right)$ be the propositional function $m \mathop \le j \mathop \le n$.

Then we have:

and:

The result follows from Sum of Summations over Overlapping Domains.