Multiplication of Dedekind Cuts: Associativity, Commutativity, Unit, Distributivity, Reciprocals and Nonnegative Products

In the setting of The Integers and the Rational Numbers, with the Natural Numbers and the Integers Identified with Subsets of the Rationals, let $C$, $\le_{C}$ and $u^{*}$ for $u\in\mathbb{Q}$ be as in Construction of the Dedekind Cuts: They Form a Set Totally Ordered by Inclusion, Closed under Sums, Negatives and Products of Nonnegative Cuts, and let $+$, $\cdot$, $0_{\mathbb{R}}$ and $1_{\mathbb{R}}$ be as in The Real Numbers §operations and The Real Numbers §constants (so that $\mathbb{R}=C$, by The Real Numbers §reals). Let $x,y,z\in C$ and $u,v\in\mathbb{Q}$.

¶ $(x\cdot y)\cdot z=x\cdot(y\cdot z)$, $x\cdot y=y\cdot x$, $x\cdot1_{\mathbb{R}}=x$ and $x\cdot(y+z)=x\cdot y+x\cdot z$.

¶ $0_{\mathbb{R}}\neq1_{\mathbb{R}}$.

¶ If $x\neq0_{\mathbb{R}}$, there is $w\in C$ with $x\cdot w=1_{\mathbb{R}}$.

¶ If $0_{\mathbb{R}}\le_{C}x$ and $0_{\mathbb{R}}\le_{C}y$, then $0_{\mathbb{R}}\le_{C}x\cdot y$.

¶ $(u\cdot v)^{*}=u^{*}\cdot v^{*}$.