Summation of Sum of Mappings on Finite Set

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

Let $S$ be a finite set.

Let $f,g : S \to \mathbb A$ be mappings.

Let $h=f+g$ be their sum.

Then we have the equality of summations on finite sets:
 * $\displaystyle \sum_{s \mathop \in S} h(s) = \sum_{s \mathop \in S} f(s) + \sum_{s \mathop \in S} g(s)$

Outline of Proof
Using the definition of summation on finite set, we reduce this to Indexed Summation of Sum of Mappings.

Proof
Let $n$ be the cardinality of $S$.

Let $\sigma : \N_{<n} \to S$ be a bijection, where $\N_{<n}$ is an initial segment of the natural numbers.

By definition of summation on finite set, we have to prove the following equality of indexed summations:
 * $\displaystyle \sum_{i \mathop = 0}^{n-1} h(\sigma(i)) = \sum_{i \mathop = 0}^{n-1} f(\sigma(i)) + \sum_{i \mathop = 0}^{n-1} g(\sigma(i))$

By Sum of Mappings Composed with Mapping, $h\circ\sigma = f\circ\sigma + g\circ \sigma$.

The above equality now follows from Indexed Summation of Sum of Mappings.

Also see

 * Indexed Summation of Sum of Mappings