Let $K$ be a \reftext{def:field-c54-2026b}{field}, let $V$ be a \reftext{def:vector-space-2026a}{vector space over $K$} with vector addition $+$, let $n$ be a \reftext{def:natural-numbers-2026a}{natural number} with successor map $S$ as in that definition, let $[n]$ be the \reftext{def:initial-segment-natural-numbers-2026a}{initial segment} determined by $n$, and let $v:[n]\to V$ be a map, whose value at $k$ is written $v_{k}$.

Let $\sigma:[n]\to V$ be the map given by \ref{lem:iterated-binary-operation-2026a} for the vector addition of $V$, that is, the unique map $[n]\to V$ with

For $j\in[n]$ the \textbf{finite sum} of $v_{1},\dots,v_{j}$ is