Ordered field/Elementary properties/3/Fact

In an ordered field, the following properties hold.

1. ${}1\geq 0$.
2. ${}a\geq 0$ holds if and only if ${}-a\leq 0$ holds.
3. ${}a\geq b$ holds if and only if ${}a-b\geq 0$ holds.
4. ${}a\geq b$ holds if and only if ${}-a\leq -b$ holds.
5. ${}a\geq b$ and ${}c\geq d$ imply ${}a+c\geq b+d$.
6. ${}a\geq b$ and ${}c\geq 0$ imply ${}ac\geq bc$.
7. ${}a\geq b$ and ${}c\leq 0$ imply ${}ac\leq bc$.
8. ${}a\geq b\geq 0$ and ${}c\geq d\geq 0$ imply ${}ac\geq bd$.
9. ${}a\geq 0$ and ${}b\leq 0$ imply ${}ab\leq 0$.
10. ${}a\leq 0$ and ${}b\leq 0$ imply ${}ab\geq 0$.

Write a proof