reciprocal in ordered field

Let $\displaystyle \mathbb{F} $ be an ordered field and $\displaystyle a \in \mathbb{F} $. If $\displaystyle a > 0 $, show that $\displaystyle a^{-1} > 0 $. If $\displaystyle a < 0 $, show that $\displaystyle a^{-1} < 0 $.

So let $\displaystyle a,b,c \in \mathbb{F} $. If $\displaystyle ab = c $ and any two of $\displaystyle a,b, \ \text{or} \ c $ is positive, then so is the third.

Take $\displaystyle b = a^{-1} $ and the Corollary is shown?

Is this correct? Generally, with corollaries, you invoke the theorem?