Equivalence of Definitions of Equivalent Division Ring Norms/Open Unit Ball Equivalent implies Norm is Power of Other Norm/Lemma 2/Lemma 2.3

Theorem
Let $R$ be a division ring.

Let $\norm{\,\cdot\,}_1: R \to \R_{\ge 0}$ and $\norm{\,\cdot\,}_2: R \to \R_{\ge 0}$ be norms on $R$.

Let $\norm{\,\cdot\,}_1$ and $\norm{\,\cdot\,}_2$ satisfy:
 * $\forall x \in R:\norm{x}_1 \lt 1 \iff \norm{x}_2 \lt 1$

Let $x_0, x \in R \setminus 0_R$ such that $\norm{x_0}_1, \norm {x}_1 \lt 1$.

Let $\alpha = \dfrac {\log \norm {x_0}_1 } {\log \norm {x_0}_2 }$ and $\beta = \dfrac {\log \norm {x}_1 } {\log \norm {x}_2 }$.

Then:
 * $\alpha = \beta$

Proof
Let $n$ and $m$ be any strictly positive integers.

Then:

By Logarithm is Strictly Increasing then:
 * \log \norm{x}_1^n \lt \log \norm{x_0}_1^m
 * \log \norm{x}_2^n \lt \log \norm{x_0}_2^m