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 $y \neq 0_R$ where $0_R$ is the zero of $R$.

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