Non-Archimedean Norm iff Non-Archimedean Metric/Necessary Condition

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

Let $d$ be the metric induced by $\norm {\,\cdot\,}$.

Then $d$ is a non-Archimedean metric.

Proof
By Metric Induced by Norm on a Normed Division Ring is Metric then $d$ satisfies the metric space axioms (M1)-(M4).

To complete the proof, all that remains is to show that $d$ is non-Archimedean.

Let $x, y, z \in R$.