Indexed Summation over Interval of Length Two

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

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

Let $f : \{a, a+1\} \to \mathbb A$ be a real-valued function.

Then the indexed summation:


 * $\displaystyle \sum_{i \mathop = a}^{a+1} f(i) = f(a) + f(a+1)$.

Proof
We have: