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

(i): the addition is a binary operation on $V$ by Vector Space over a Field, so $u+v\in V$ by Binary Operations on a Set (ii).

(ii): the scalar multiplication is a function from $\mathbb{R}\times V$ to $V$ by Vector Space over a Field, and $(\lambda,v)\in\mathbb{R}\times V$, so its value $\lambda v$ lies in its range, which is contained in $V$ by Functions, Values of a Function, and Functions from One Class to Another (iii).

(iii): since $0_{V}\in V$ and $x\in V$, Vector Space over a Field (ii) and The Zero Vector, Negatives and Differences in a Vector Space (i) give $0_{V}+x=x+0_{V}=x$.