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 Normed Division Ring is Metric then $d$ satisfies the metric space axioms $(\text M 1)$ to $(\text M 4)$.

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

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