(related to Proposition: $0$ Is Less Than $1$ In Ordered Fields)

- Let $(F,+,\cdot)$ be an ordered field.
- Let $0\in F$ be the zero element of addition in the field $F$, and let $1\in F$ be the unity element of multiplication in $F$.
- It has been shown that $1\neq 0$ in a field.
- It has also been shown that for all $x\in F$ with $x\neq 0$ we have $x^2 > 0$.
- Thus, it follows $1^2 > 0.$
- Because $1$ is the unity element of multiplication in $F$, we also have $1\cdot 1=1^2=1.$
- It follows $1 > 0.$∎

**Knauer Ulrich**: "Diskrete Strukturen - kurz gefasst", Spektrum Akademischer Verlag, 2001