Finite Sum Notation in a Vector Space

Statement

Let $K$ be a field, let $V$ be a vector space over $K$ with vector addition $+$ , let $n$ be a natural number with successor map $S$ as in that definition, let $[n]$ be the 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 Existence and Uniqueness of Iterates of a Binary Operation for the vector addition of $V$ , that is, the unique map $[n]\to V$ with

\sigma(1)=v_{1},\qquad \sigma(S(m))=\sigma(m)+v_{S(m)}\quad\text{whenever }S(m)\in[n].

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

\sum_{k=1}^{j}v_{k}=\sigma(j).

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