Properties of Norm on Division Ring/Norm of Quotient

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 $x, y \in R$

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

Proof
Let $y \ne 0_R$.

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

So:

Similarly:
 * $\norm{y^{-1}x} = \dfrac {\norm{x}}{\norm{y}}$