Uniqueness of the Zero Vector and of Negatives

In the setting of Routine Facts Used Without Citation: Arithmetic, Maps, Predicativity and Indices, let $K$, with $+$, $\cdot$, $0$ and $1$, be a field, and let $V$, with its addition and scalar multiplication, be a vector space over $K$.

(i) The zero vector is unique. There is exactly one $z\in V$ such that $v+z=v$ for every $v\in V$.

(ii) Negatives are unique. Let $z$ be the element of $V$ given by (i). For every $v\in V$ there is exactly one $w\in V$ with $v+w=z$.