Ordered field/Elementary properties for inverses/Fact

In an ordered field, the following properties holds.

1. From ${}x>0$ one can deduce ${}x^{-1}>0$.
2. From ${}x<0$ one can deduce ${}x^{-1}<0$.
3. For ${}x>0$ we have ${}x\geq 1$ if and only if ${}x^{-1}\leq 1$.
4. From ${}x\geq y>0$ one can deduce ${}x^{-1}\leq y^{-1}$.
5. For positive elements ${}x,y$ the relation ${}x\geq y$ is equivalent with ${}{\frac {x}{y}}\geq 1$.

Proof, Write another proof