Cancelling an Added Real, and the Product of a Real with a Negative

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 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); for $x,y\in\mathbb{R}$, $-y$ is the negative of $y$ and $x-y$ is the difference of $x$ and $y$. Let $x$, $y$ and $z$ be real numbers.

(i) If $x+z=y+z$, then $x=y$.

(ii) $x\cdot(-y)=-(x\cdot y)$.