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Proof

(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 $1 > 0.$
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References

Bibliography

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