The Norm of a Negative, and the Norm Distance of a Translate

In the setting of Routine Facts, Continued: Typed Arithmetic, Maps and Predicativity and Real Normed, Inner Product and Hilbert Spaces: Notation:

(i) Norm of a negative. Let $V$ be a real vector space and $\lVert\cdot\rVert$ a norm on $V$. Then for every $v\in V$

$$\lVert -v\rVert=\lVert v\rVert .$$

(ii) Distance to a translate. Let $(V,\lVert\cdot\rVert)$ be a real normed space with norm metric $d_V$. Then for all $a,w\in V$

$$d_V(a+w,a)=\lVert w\rVert .$$