# Real Number is Greater than Zero iff its Negative is Less than Zero

## Theorem

$\forall x \in \R: x > 0 \iff \left({-x}\right) < 0$

## Proof

Let $x > 0$.

 $\displaystyle x$ $>$ $\displaystyle 0$ $\displaystyle \implies \ \$ $\displaystyle x + \left({-x}\right)$ $>$ $\displaystyle 0 + \left({-x}\right)$ Real Number Axioms: $\R O2$: compatibility with addition $\displaystyle \implies \ \$ $\displaystyle 0$ $>$ $\displaystyle 0 + \left({-x}\right)$ Real Number Axioms: $\R A4$: Inverses $\displaystyle \implies \ \$ $\displaystyle 0$ $>$ $\displaystyle \left({-x}\right)$ Real Number Axioms: $\R A3$: Identity

$\Box$

Let $\left({-x}\right) < 0$.

 $\displaystyle \left({-x}\right)$ $<$ $\displaystyle 0$ $\displaystyle \implies \ \$ $\displaystyle x + \left({-x}\right)$ $<$ $\displaystyle x + 0$ Real Number Axioms: $\R O2$: compatibility with addition $\displaystyle \implies \ \$ $\displaystyle 0$ $<$ $\displaystyle x + 0$ Real Number Axioms: $\R A4$: Inverses $\displaystyle \implies \ \$ $\displaystyle 0$ $<$ $\displaystyle x$ Real Number Axioms: $\R A3$: Identity

$\blacksquare$