Square of Non-Zero Real Number is Strictly Positive

Theorem

 * $\forall x \in \R: x \ne 0 \implies x^2 > 0$

Proof
There are two cases to consider:
 * $(1): \quad x > 0$
 * $(2): \quad x < 0$

Let $x > 0$.

Then:

Let $x < 0$.

Then: