A Sum over a Finite Index Set Does Not Depend on the Enumeration

Let $K$ be a field, let $F$ be a nonempty finite set, and let $n$ be a natural number such that $F$ has $n$ elements; such an $n$ is unique by Uniqueness of the Number of Elements. Let $[n]$ be the initial segment determined by $n$, and let $f:F\to K$ be a map. Sums below are the finite sums of $K$.

Then for all bijections $\varphi:[n]\to F$ and $\psi:[n]\to F$,

$$\sum_{k=1}^{n}f\bigl(\varphi(k)\bigr)=\sum_{k=1}^{n}f\bigl(\psi(k)\bigr).$$