# 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:

 $\displaystyle x \times x$ $>$ $\displaystyle 0$ Product of Strictly Positive Real Numbers is Strictly Positive

Let $x < 0$.

Then:

 $\displaystyle x$ $<$ $\displaystyle 0$ $\displaystyle \leadsto \ \$ $\displaystyle x \times x$ $>$ $\displaystyle x \times 0$ Order of Real Numbers is Dual of Order Multiplied by Negative Number $\displaystyle$ $=$ $\displaystyle 0$ Real Zero is Zero Element

$\blacksquare$