Order is Preserved on Positive Reals by Squaring/Proof 3

Theorem
Let $x, y \in \R: x > 0, y >0$.

Then:
 * $x < y \iff x^2 < y^2$

Proof
From Real Numbers form Totally Ordered Field, the real numbers form a totally ordered field.

By definition, a totally ordered field is a totally ordered ring without proper zero divisors.

The result follows from Order of Squares in Totally Ordered Ring without Proper Zero Divisors.