Subring of Non-Archimedean Division Ring

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

Let $\struct {S, \norm {\, \cdot \,}_S }$ be a normed division subring of $R$.

Then:
 * $\norm {\, \cdot \,}_S$ is a non-archimedean norm.

Proof
$\forall x, y \in S$: