Indexed Summation over Adjacent Intervals

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

Let $a,b,c$ be integers.

Let $\left[{a \,.\,.\, c}\right]$ denote the integer interval between $a$ and $b$.

Let $b \in \left[{a-1 \,.\,.\, c}\right]$.

Let $f : \left[{a \,.\,.\, c}\right] \to \mathbb A$ be a mapping.

Then we have an equality of indexed summations:


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