Let $(F,+,*,{\preceq})$ be a totally ordered field.
Then ${\forall}f\,{\in}\,F:0_F\,{\preceq}\,f*f$.
It's obvious from the definition of total ordering on a field that this applies to $f:0_F\,{\preceq}\,f$. Since $0_F\,{\preceq}\,a\,{\land}\,0_F\,{\preceq}\,b{\implies}0_F\,{\preceq}\,a*b$, we know that $f^2$ must succeed $0_F$.
What about the cases $f:f\,{\preceq}\,0_F\,{\land}\,f\,{\neq}\,0_F$? How do we prove the theorem for them?