Properties of Norm on Division Ring/Norm of Power Equals Unity

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:
 * $\forall n \in \N_{\gt 0}: \norm{x^{n}} = 1 \implies {\norm{x}} = 1$

Proof
Let $n \in \N_{\gt 0}$.

Let $\norm{x^{n}} = 1$.

By Norm axiom (N2) (Multiplicativity) then:
 * $\norm{x}^{n} = 1$

Since $\norm {x} \ge 0$, by Positive Real Complex Root of Unity then:
 * ${\norm{x}} = 1$

as desired.