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

Proof
Let $y \ne 0_R$.

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

So:

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