Ordered field/Detailed/Definition

A field ${\displaystyle {}K}$ is called an ordered field, if there is a relation ${\displaystyle {}>}$ (larger than) between the elements of ${\displaystyle {}K}$, fulfilling the following properties (${\displaystyle {}a\geq b}$ means ${\displaystyle {}a>b}$ or ${\displaystyle {}a=b}$).

1. For two elements ${\displaystyle {}a,b\in K}$, we have either ${\displaystyle {}a>b}$ or ${\displaystyle {}a=b}$ or ${\displaystyle {}b>a}$.
2. From ${\displaystyle {}a\geq b}$ and ${\displaystyle {}b\geq c}$, one may deduce ${\displaystyle {}a\geq c}$ (for any ${\displaystyle {}a,b,c\in K}$).
3. ${\displaystyle {}a\geq b}$ implies ${\displaystyle {}a+c\geq b+c}$ (for any ${\displaystyle {}a,b,c\in K}$).
4. From ${\displaystyle {}a\geq 0}$ and ${\displaystyle {}b\geq 0}$, one may deduce ${\displaystyle {}ab\geq 0}$ (for any ${\displaystyle {}a,b\in K}$).