The Negative of Zero in ℝ, and Multiplying an Inequality on the Left by a Nonnegative Real

This lemma 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 (ii). By The Structure of ℝ as a Field and an Ordered Ring (i), $\mathbb{R}$, with $+$, $\cdot$, $0$ and $1$, is a field, hence a commutative ring, so for $x\in\mathbb{R}$ its negative $-x$ is defined.

(i) $-0=0$.

(ii) For all real numbers $x$, $y$ and $z$: if $x\le y$ and $0\le z$, then $z\cdot x\le z\cdot y$.