Characterisation of Non-Archimedean Division Ring Norms/Sufficient Condition/Lemma 1

Theorem
Let $\struct {R, \norm {\,\cdot\,} }$ be a normed division ring with unity $1_R$.

Let $x, y \in R$. Let $y \ne 0_R$ where $0_R$ is the zero of $R$.

Then:
 * $\norm {x + y} \le \max \set {\norm x, \norm y} \iff \norm {x y^{-1} + 1_R} \le \max \set {\norm {x y^{-1} }, 1}$