Closure Facts in an Ordered Field: Negatives, Differences, Quotients, Absolute Values, Powers, Natural Numbers, Halves and Minima

In the setting of Commutative Rings, Fields and Ordered Fields: Standard Notation, let $F$, with $+$, $\cdot$, $0$, $1$ and a total order $\le$, be an ordered field, and let $x,y\in F$.

¶ The negative $-x$ and the difference $x-y=x+(-y)$ lie in $F$.

¶ If $y\neq0$, then the reciprocal $y^{-1}$ and the quotient $x/y=x\cdot y^{-1}$ lie in $F$.

¶ The absolute value $|x|$ lies in $F$.

¶ For every $n\in\mathbb{N}_{0}$, the power $x^{n}$ lies in $F$, with $x^{0}$ as in Powers with Exponents in the Natural Numbers with Zero §zero.

¶ For every $n\in\mathbb{N}_{0}$, the image $n_{F}$ of $n$ in $F$ lies in $F$.

¶ The element $2=2_{F}$, read as in Commutative Rings, Fields and Ordered Fields: Standard Notation §numerals, lies in $F$ and satisfies $2=1+1$ and $2\neq0$ in $F$.

¶ With $2$ as in the clause two, $x/2\in F$ and $x/2+x/2=x$ in $F$.

¶ Let $a,b\in F$ with $0<a$ and $0<b$, and let $\min\{a,b\}$ be as in The Maximum and Minimum of Two Elements of a Total Order §exists. Then $\min\{a,b\}\in F$, $0<\min\{a,b\}$, $\min\{a,b\}\le a$ and $\min\{a,b\}\le b$; in particular, there is $c\in F$ with $0<c$, $c\le a$ and $c\le b$.