Isometrically Isomorphic Non-Archimedean Division Rings

Theorem
Let $\struct {R, \norm {\,\cdot\,}_R}$ and $\struct {S, \norm {\,\cdot\,}_S}$ be normed division rings.

Let $\phi:R \to S$ be an isometric isomorphism.

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

Necessary Condition
Let $\norm {\,\cdot\,}_R$ be a non-archimedean norm.

Then for all $x,y \in R$:

Sufficient Condition
Let $\norm {\, \cdot \,}_S$ be a non-archimedean norm.

By Inverse of Isometric Isomorphism, $\phi^{-1}: S \to R$ is an isometric isomorphism.

By the necessary condition, $\norm {\, \cdot \,}_R$ is non-archimedean.