Guided Example #2 - Algebraic Structures

(indirect proof) Given ${\displaystyle a>0}$, assume temporarily ${\displaystyle -a\ge 0}$.

Then ${\displaystyle -a\ge 0}$ and ${\displaystyle -a+a\ge 0+a}$ (Addition property of order)

Then ${\displaystyle 0\ge 0+a}$. (additive inverse)(additive identity).

But this contradicts the hypothesis. Therefore, ${\displaystyle \mathrm{-a}<0.}$