Triangle Inequality/Real Numbers/Proof 5

Proof
From Negative of Absolute Value, it is sufficient to prove that:
 * $\size x + \size y \ge x + y$

and:
 * $\size x + \size y \ge -\paren {x + y}$

By definition of absolute value:
 * $x \le \size x$

and:
 * $y \le \size y$

Then:
 * $x + y \le \size x + \size y$

We also have that:

Hence the result.