Division Subring of Normed Division Ring

Theorem
Let $\struct {R, \norm {\, \cdot \,} }$ be a normed division ring.

Let $S$ be a division subring of $R$.

Then:
 * $\struct {S, \norm {\, \cdot \,}_S }$ is a normed division subring of $\struct {R, \norm {\, \cdot \,} }$

where $\norm {\, \cdot \,}_S$ is the norm $\norm{\,\cdot\,}$ restricted to $S$.

(N1) : Positive Definiteness
$\forall x \in S$:

(N2) : Multiplicativity
$\forall x, y \in S$

(N3) : Triangle Inequality
$\forall x, y \in S$