A Finite Sum of Vectors with Vanishing Tail

Let $K$ be a \reftext{def:field-c54-2026b}{field}, let $V$ be a \reftext{def:vector-space-2026a}{vector space over $K$} with \reftext{lem:vector-space-basic-identities-2026a}{zero vector} $0_{V}$, let $n$ be a \reftext{def:natural-numbers-2026a}{natural number}, let $[n]$ be the \reftext{def:initial-segment-natural-numbers-2026a}{initial segment} it determines, and let $<$ be the \reftext{def:order-natural-numbers-2026a}{strict order} on $\mathbb{N}$.

Let $u\in V^{n}$ be an \reftext{def:finite-tuple-power-2026a}{$n$-tuple} in $V$ and let $j\in[n]$ be such that $u_{k}=0_{V}$ for every $k\in[n]$ with $j<k$. Then, with \reftext{def:finite-sum-vector-space-2026a}{finite sums in $V$},