The Iterated Operation over a Finite Set Does Not Depend on the Enumeration

In the setting of The Natural Numbers and the Natural Numbers with Zero: Arithmetic, Order, Induction and Recursion, let $\ast$ be an associative and commutative binary operation on a set $X$, let $A$ be a nonempty finite set with $n=\#A$ as in The Number of Elements of a Finite Set §cardinality, and let $f:A\to X$. Let $[n]$ be as in Intervals of Natural Numbers §segment and iterated operations as in Iterated Operations: Finite Sums and Finite Products §iterated.

¶ $n\in\mathbb{N}$.

¶ For all bijections $\varphi,\psi$ from $[n]$ onto $A$,

$$\mathop{\ast}\limits_{k=1}^{n}f(\varphi(k))=\mathop{\ast}\limits_{k=1}^{n}f(\psi(k)).$$