Indexed Summation over Interval of Length One

Theorem
Let $\mathbb A$ be one of the standard number systems $\N, \Z, \Q, \R, \C$.

Let $a \in \Z$ be an integer.

Let $f: \set a \to \mathbb A$ be a mapping on the singleton $\set a$.

Then the indexed summation:


 * $\ds \sum_{i \mathop = a}^a \map f i = \map f a$

Proof
We have:

Also see

 * Indexed Summation over Interval of Length Two
 * Summation over Singleton Set