Membership and the Zero Law of a Real Vector Space in Flat Form

In the setting of Routine Facts, Continued: Typed Arithmetic, Maps and Predicativity and Real Normed, Inner Product and Hilbert Spaces: Notation, let $V$, with its addition and scalar multiplication, be a real vector space.

(i) Closure under addition. If $u,v\in V$, then $u+v\in V$.

(ii) Closure under scalar multiplication. If $\lambda\in\mathbb{R}$ and $v\in V$, then $\lambda v\in V$.

(iii) Zero on the left. If $x\in V$, then $0_{V}+x=x$.