Reverse Triangle Inequality/Normed Division Ring

Theorem
Let $\struct {R, \norm {\,\cdot\,} }$ be a normed division ring with zero: $0$.

Then:
 * $\forall x, y \in R: \norm {x - y} \ge \big\lvert {\norm x - \norm y} \big\rvert$

Proof
Let $d$ denote the metric induced by $\norm {\, \cdot \,}$, that is,
 * $d \tuple {x, y} = \norm {x - y}$

From Metric Induced by Norm is Metric we have that $d$ is indeed a metric.

Then, from the Reverse Triangle Inequality as applied to metric spaces:
 * $\forall x, y, z \in R: \big\lvert {\norm {x - z} - \norm {y - z}} \big\rvert \le \norm {x - y}$

Then:
 * $\forall x, y \in R: \big\lvert{\norm x - \norm y}\big\rvert = \big\lvert{\norm{x - 0} - \norm{y - 0} }\big\rvert \le \norm {x - y}$.