Difference and Quotient Identities in ℝ

In the setting of Routine Facts Used Without Citation: Arithmetic, Maps, Predicativity and Indices, let $x,y,h,a,b,c,m\in\mathbb{R}$. Negatives and differences are as in Negatives, Differences, Reciprocals and Quotients (i), reciprocals and quotients as in Negatives, Differences, Reciprocals and Quotients (ii), absolute values as in Absolute Value in an Ordered Field (i), and $<$ is the strict relation of the order $\le$ of $\mathbb{R}$.

(i) $x-y=x+(-y)$; $(x+h)-x=h$; $x-(x-h)=h$; $x-x=0$; $-(x-y)=y-x$; and $x<y$ if and only if $0<y-x$.

(ii) If $b\neq0$, then $x/b=x\cdot b^{-1}$; $(x/b)\cdot b=x$ and $b\cdot(x/b)=x$; $(a-m\cdot b)/b=a/b-m$; and $(a-c)/b=a/b-c/b$.

(iii) If $0<x$ and $0<b$, then $0<x/b$. If $b<0$, then $b\neq0$ and $b^{-1}<0$. If $b\neq0$, then $|b|\neq0$, $|b^{-1}|=|b|^{-1}$ and $|x/b|=|x|/|b|$.