Proof of Negatives, Reciprocals and the Strict Order in an Ordered Field, as Values

Proof of (i) and (ii). By Ordered Fields (i), $F$ is a field, in particular a commutative ring. By Negatives, Differences, Reciprocals and Quotients (i), $-x\in F$ and $x+(-x)=0$; by Negatives, Differences, Reciprocals and Quotients (ii), if $x\neq0$, then $x^{-1}\in F$ and $x\cdot x^{-1}=1$.

Proof of (iii). The order $\le$ is a partial order on $F$, by Ordered Fields (i). By Uniqueness of Least and Greatest Elements, Properties of the Strict Order, and Trichotomy for Total Orders (ii), $x<y$ if and only if $x\le y$ and $x\neq y$. If $x\le y$ and $x\neq y$, then $y\le x$ would give $x=y$ by antisymmetry, so $y\le x$ fails. Conversely, if $x\le y$ and $y\le x$ fails, then $x\neq y$, since $x=y$ would turn $x\le y$ into $y\le x$. Hence $x<y$ if and only if $x\le y$ and not $y\le x$.