Equivalence of Definitions of Equivalent Division Ring Norms/Open Unit Ball Equivalent implies Norm is Power of Other Norm/Lemma 2/Lemma 2.2

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:
 * $\forall y \in R:\norm{y}_1 \lt 1 \iff \norm{y}_2 \lt 1$

Then:
 * $\forall y \in R:\norm{y}_1 = 1 \iff \norm{y}_2 = 1$

Proof
By assumption then:
 * $\forall y \in R:\norm{y}_1 \ge 1 \iff \norm{y}_2 \ge 1$

By Lemma 1 then:
 * $\forall y \in R:\norm{y}_1 \le 1 \iff \norm{y}_2 \le 1$

Hence $\forall y \in R$: