The Maximum and Minimum of Two Elements of an Ordered Field: Sum, Difference and Absolute Value

In the setting of Commutative Rings, Fields and Ordered Fields: Standard Notation, let $F$, with a total order $\le$, be an ordered field, with $\max$ and $\min$ of two elements as in The Maximum and Minimum of Two Elements of a Total Order §exists, and let $a,b\in F$.

¶ $\max\{a,b\}+\min\{a,b\}=a+b$.

¶ $\max\{a,b\}-\min\{a,b\}=|a-b|$.

¶ $|a|=\max\{a,-a\}$.