Summation over Finite Index is Well-Defined

Theorem
Let $\struct{G, +}$ be a commutative monoid.

Let $\family{g }_{i \mathop \in I}$ be an indexed subset of $G$ where the indexing set $I$ is finite.

Then the summation $\ds \sum_{i \mathop \in I} g_i$ is well-defined.

Proof
To show that summation over $I$ is well-defined it needs to be shown:
 * $(1) \quad \exists$ a finite enumeration of $I$
 * $(2) \quad \forall$ finite enumerations $i$ and $j$ of $I : \ds \sum_{k \mathop = 1}^n g_{i_k} = \sum_{k \mathop = 1}^n g_{j_k}$

Proof of $(1)$
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.

Proof of $(2)$
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.