Given an abelian additive group $G$ and a sequence $(a_n)_{n\in\mathbb N}$ of elements of $G$, we're able to define recursively "indexed summation" through : $$ \sum_{i=0}^0 a_i = a_0 \;\;\;\text{ and }\;\;\; \sum_{i=0}^{n+1}a_i = \sum_{i=0}^{n}a_i + a_{n+1}$$ Then, a set $I$ is said to be finite iff there exists a non-zero positive integer $n$ such that, there exists a bijection $\sigma$ between $I$ and $[1, n] = \{1,...,n\}$. Now, considering a family $(a_i)_{i\in I}$ of elements of $G$ and a finite subset $J\subseteq I$, the most natural way to define summation over finite sets is : $$ \sum_{i\in J} a_i = \sum_{i=0}^{n} a_{\sigma(i)} $$ where $\sigma$ denotes a bijection between $J$ and the integer interval $[1,n+1]$. However, this sum must be well-defined, it means that it holds for any bijection between those sets.
How to prove that this summation is well-defined (without using the symmetric group $S_n$) ?
Thanks for answers :)