Order is Preserved on Positive Reals by Squaring/Proof 4

Necessary Condition
Let $x < y$.

Then:

So:
 * $x < y \implies x^2 < y^2$

Sufficient Condition
Let $x^2 < y^2$.

$x \ge y$.

Then:

But this contradicts our assertion that $x^2 < y^2$.

Hence by Proof by Contradiction it follows that:
 * $x < y$