Rules of Order in an Ordered Ring: Differences, Sums, Negatives, Products and Squares

In the setting of Sets and Maps: Ordinary Notation, let $R$, with $+$, $\cdot$, $0$, $1$ and a total order $\le$ with strict relation $<$, be an ordered ring, with negatives and differences as in Negatives, Differences, Reciprocals and Quotients §negative. Let $x,y,z,w\in R$.

¶ $x\le y$ if and only if $0\le y-x$, and $x<y$ if and only if $0<y-x$.

¶ $x\le y$ if and only if $x+z\le y+z$, and $x<y$ if and only if $x+z<y+z$. If $x\le y$ and $z\le w$, then $x+z\le y+w$; if moreover $x<y$, then $x+z<y+w$.

¶ $x\le y$ if and only if $-y\le-x$, and $x<y$ if and only if $-y<-x$.

¶ If $x\le y$ and $0\le z$, then $x\cdot z\le y\cdot z$; if $x\le y$ and $z\le0$, then $y\cdot z\le x\cdot z$.

¶ If $x\le0\le y$, then $x\cdot y\le0$; if $x\le0$ and $y\le0$, then $0\le x\cdot y$.

¶ $0\le x\cdot x$ and $0\le1$; if $0\neq1$, then $0<1$.

¶ If $0\le x\le y$ and $0\le z\le w$, then $x\cdot z\le y\cdot w$.