Reverse Triangle Inequality/Real and Complex Fields/Proof 1

Proof
Let $X$ denote either $\R$ or $\C$ as appropriate.

From Real Number Line is Metric Space and Complex Plane is Metric Space the distance function $d: X \times X \to \R$ can be defined as:
 * $\map d {x, y} = \size {x - y}$

From the Reverse Triangle Inequality as applied to metric spaces:
 * $(1): \quad \forall x, y, z \in X: \size {\map d {x, z} - \map d {y, z} } \le \map d {x, y}$

Let $z = 0$.

Then $(1)$ translates to:
 * $\forall x, y, z \in X: \size {\size {x - 0} - \size {y - 0} } \le \size {x - y}$

Hence the result.