Summation over Finite Subset is Well-Defined

Theorem
Let $G$ be a commutative monoid.

Let $F \subseteq G$ be a finite subset of $G$.

Then the summation over $F$ is well-defined.

Proof
By definition of finite set:
 * $\exists n \in \N : \exists$ a bijection $g: \closedint 1 n \to F$

Hence $g$ is a finite enumeration of $F$ by definition.

So the summation $\ds \sum_1^n g_i$ exists.

Let $h: \closedint 1 n \to F$ be any other finite enumeration of $F$.

Consider the composite mapping $g^{-1} \circ h : \closedint 1 n \to \closedint 1 n$ which exists and is a bijection because $g$ and $h$ are bijections.

Let $\iota: G \to G$ denote the identity mapping on $G$.

We have:

The result follows.