Properties of Norm on Division Ring/Norm of Unity

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

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

Then:
 * $\norm {1_R} = 1$.

Proof
By the norm axiom (N2) (Multiplicity) then:
 * $\forall x, y \in R: \norm {x \circ y} = \norm{x} \norm{y}$

In particular:
 * $\norm{1_R} = \norm{1_R \circ 1_R} = \norm{1_R} \norm{1_R}$

By the norm axiom (N1) (Positive defintiteness) then:
 * $\norm{1_R} \ne 0$

So $\norm{1_R}$ has an inverse in $\R$.

By multiplying by this inverse, then:
 * $ \norm{1_R} \norm{1_R} =\norm{1_R} \iff \norm{1_R} = 1$

as desired.