Properties of Norm on Division Ring/Norm of Inverse

Theorem
Let $\struct {R, +, \circ}$ be a division ring with zero $0_R$ and unity $1_R$.

Let $\norm {\, \cdot \,}$ be a norm on $R$.

Let $x \in R$

Then:
 * $x \ne 0_R \implies \norm {x^{-1} } = \dfrac 1 {\norm x}$

Proof
Let $x \ne 0_R$.

By Norm axiom (N1) (Positive Definiteness) then:
 * $\norm x \ne 0$

So: