Invariance of Finite Sums and Products under Reindexing by a Permutation

Let $K$ be a field. Let $\mathbb{N}$ be the set of natural numbers, let $n\in\mathbb{N}$, and let $[n]$ be the initial segment determined by $n$. Let $a:[n]\to K$ be a map with values written $a_{k}$, and let $\sigma:[n]\to[n]$ be a bijection. Sums and products below are the finite sums and the finite products of $K$.

Then the following hold.

1. (Sums)

2. (Products)