The Structure of ℝ as a Field and an Ordered Ring

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.

¶ $\mathbb{R}$, with $+$, $\cdot$, $0$ and $1$, is a field.

¶ $\mathbb{R}$, with $+$, $\cdot$, $0$, $1$ and $\le$, is an ordered ring.

¶ $\le$ is a partial order on $\mathbb{R}$.

¶ The binary operation $\cdot$ on $\mathbb{R}$ is associative.

¶ The binary operation $\cdot$ on $\mathbb{R}$ is commutative.

¶ $0$ is a neutral element of the binary operation $+$ on $\mathbb{R}$.

¶ For all real numbers $x$ and $y$: if $x<y$, where $<$ is the strict relation associated with $\le$, then $x\le y$.

¶ For all real numbers $x$ and $y$, $(x-y)+y=x$, where $x-y$ is the difference of $x$ and $y$.

¶ For every real number $x$, $x^{2}=x\cdot x$, where $x^{2}$ is the power of $x$ for $\cdot$ with exponent the digit $2$, read in $\mathbb{N}_{0}$ as an exponent by Commutative Rings, Fields and Ordered Fields: Standard Notation §numerals.