Concatenation of Finite Sums

Let $K$ be a field. Let $\mathbb{N}$ be the set of natural numbers, with addition $+$ and successor map $S$ as in that definition, and for a natural number $p$ let $[p]$ be the initial segment determined by $p$, that is, the set of natural numbers $k$ with $1\le k\le p$.

Let $m,n\in\mathbb{N}$ and let $a\in K^{n+m}$ be an $(n+m)$-tuple in $K$, with components $a_{k}$; by that definition $a$ is a map from $[n+m]$ to $K$. Then $[n]\subseteq[n+m]$, and $n+k\in[n+m]$ for every $k\in[m]$, by claims 4 and 6 of Properties of the Order on the Natural Numbers. Let $a'\in K^{n}$ be the restriction of $a$ to $[n]$, and let $b\in K^{m}$ be the $m$-tuple with components $b_{k}=a_{n+k}$.

All sums below are finite sums in a field. Then