Inequalities in an Ordered Field: Mixed Transitivity, Strict Sums, Signs, Products, Natural Numbers, Halving, Reciprocals, Absolute Values and Squares

In the setting of Commutative Rings, Fields and Ordered Fields: Standard Notation, let $F$ be an ordered field, and let $x,y,z,w\in F$.

¶ If $x\le y$ and $y<z$, or $x<y$ and $y\le z$, then $x<z$.

¶ If $x<y$ and $z\le w$, then $x+z<y+w$.

¶ $x<y$ if and only if $-y<-x$.

¶ If $0<x$ and $0<y$, then $0<xy$; if $0\le x$ and $0\le y$, then $0\le xy$.

¶ If $x\le y$ and $0\le z$, then $xz\le yz$.

¶ In $F$, $0\le n$ for every $n\in\mathbb{N}_{0}$, and $0<n$ for every $n\in\mathbb{N}$.

¶ $0<2$ in $F$, and if $0<x$, then $0<x/2<x$ and $x/2+x/2=x$.

¶ If $0<x\le y$, then $0<y^{-1}\le x^{-1}$.

¶ $|x|<y$ if and only if $-y<x$ and $x<y$.

¶ $\big||x|-|y|\big|\le|x-y|$.

¶ $|x-y|\le|x-z|+|z-y|$.

¶ $|x|^{2}=x^{2}$.

¶ $(x-y)^{2}\le2\,(x^{2}+y^{2})$.