Reverse Triangle Inequality/Real and Complex Fields/Proof 2

Theorem
Let $x$ and $y$ be elements of either the real numbers $\R$ or the complex numbers $\C$.

Then:
 * $\left\vert{x - y}\right\vert \ge \left\vert{\left\vert{x}\right\vert - \left\vert{y}\right\vert}\right\vert$

where $\left\vert{x}\right\vert$ denotes either the absolute value of a real number or the complex modulus of a complex number.

Proof
From the proof 2 of the corollary to this result, which is derived independently:
 * $\left\vert{x - y}\right\vert \ge \left\vert{x}\right\vert - \left\vert{y}\right\vert$

There are two options:

$(1): \quad$ If $\left\vert{x}\right\vert \ge \left\vert{y}\right\vert$:
 * $\left\vert{ \left\vert{x}\right\vert - \left\vert{y}\right\vert }\right\vert = \left\vert{x}\right\vert - \left\vert{y}\right\vert$, and the proof is finished.

$(2): \quad$ If $\left\vert{y}\right\vert \ge \left\vert{x}\right\vert$:
 * $\left\vert{y - x}\right\vert \ge \left\vert{y}\right\vert - \left\vert{x}\right\vert = \left\vert{ \left\vert{y}\right\vert - \left\vert{x}\right\vert }\right\vert$

But:
 * $\left\vert{y - x}\right\vert = \left\vert{x - y}\right\vert$

and:
 * $\left\vert{ \left\vert{y}\right\vert - \left\vert{x}\right\vert }\right\vert = \left\vert{ \left\vert{x}\right\vert - \left\vert{y}\right\vert }\right\vert$

From this we have:
 * $-\left\vert{ \left\vert{x}\right\vert - \left\vert{y}\right\vert }\right\vert \ge -\left\vert{x - y}\right\vert$

Since, by Negative of Absolute Value, we have that:
 * $\left\vert{x}\right\vert - \left\vert{y}\right\vert \ge -\left\vert{ \left\vert{x}\right\vert - \left\vert{y}\right\vert }\right\vert$

it follows that:
 * $-\left\vert{x - y}\right\vert \le \left\vert{x}\right\vert - \left\vert{y}\right\vert \le \left\vert{x - y}\right\vert$

The result follows.