Equivalence of Definitions of Non-Archimedean Division Ring Norm

Theorem
Let $\struct {R, +, \circ}$ be a division ring whose zero is denoted $0_R$.

Definition 1 implies Definition 2
Let $\norm{\,\cdot\,} : R \to \R_{\ge 0}$ be a norm on a division ring satisfying:

It remains only to show that $\norm{\,\cdot\,}$ satisfies $(N1)$ and $(N2)$.

This follows from the definition of a norm on a division ring.

Definition 2 implies Definition 1
Let $\norm{\,\cdot\,} : R \to \R_{\ge 0}$ satisfy the non-Archimedean norm axioms: $(N1)$, $(N2)$ and $(N4)$.

To show that $\norm{\,\cdot\,}$ is a norm on a division ring satisfying $(N4)$, it remains to show that $\norm{\,\cdot\,}$ satisfies:

Let $x, y \in R$.

, suppose $\norm x \le \norm y$.

From non-Archimedean norm axiom $(N1)$ : Positive Definiteness:
 * $0 \le \norm x$

Then:

The result follows.