Generalized Distributivity: Expanding a Product of Finite Sums

Let $K$ be a field. Let $\mathbb{N}$ be the set of natural numbers with successor map $S$ as in that definition, let $m,n\in\mathbb{N}$, and let $[m]$ and $[n]$ be the initial segments they determine. Let $c$ be an $n$-tuple of $m$-tuples in $K$, with components written $c_{ik}$ for $i\in[n]$ and $k\in[m]$.

Let $[m]^{n}$ denote the set of $n$-tuples in $[m]$, that is, the set of maps $f:[n]\to[m]$; it is nonempty and finite by claim 3 of Finiteness of Cartesian Products, Tuple Sets, and Permutation Sets. Sums and products with a numerical index range are the finite sums and the finite products of $K$, and the sum indexed by $[m]^{n}$ is the sum over a finite index set of Sum over a Finite Index Set.

Then