Arithmetic and Order Rules in the Real Numbers, Stated for ℝ

This helper adopts the setting of The Real Numbers, with the Natural Numbers, Integers and Rationals Identified with Subsets of the Reals, and Completeness: $\mathbb{R}$, its order $\le$, its operations $+$ and $\cdot$, and its elements $0$ and $1$ are as in The Real Numbers, with the Natural Numbers, Integers and Rationals Identified with Subsets of the Reals, and Completeness §reals; for $x\in\mathbb{R}$, $-x$ is the negative of $x$ and $|x|$ its absolute value. Let $x$, $y$ and $z$ be real numbers.

¶ $+$ and $\cdot$ are binary operations on $\mathbb{R}$, and $x+y\in\mathbb{R}$, $x\cdot y\in\mathbb{R}$, $-x\in\mathbb{R}$, $0\in\mathbb{R}$ and $1\in\mathbb{R}$.

¶ $x+y=y+x$ and $x\cdot y=y\cdot x$.

¶ $(x+y)+z=x+(y+z)$ and $(x\cdot y)\cdot z=x\cdot(y\cdot z)$.

¶ $x\cdot(y+z)=x\cdot y+x\cdot z$ and $(x+y)\cdot z=x\cdot z+y\cdot z$.

¶ $x+0=x$, $0+x=x$, $x\cdot1=x$ and $1\cdot x=x$.

¶ $x+(-x)=0$ and $(-x)+x=0$.

¶ $0\cdot x=0$ and $x\cdot0=0$.

¶ $x\le x$; if $x\le y$ and $y\le x$, then $x=y$; if $x\le y$ and $y\le z$, then $x\le z$; and $x\le y$ or $y\le x$.

¶ $|x|\in\mathbb{R}$ and $0\le|x|$.