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

Throughout, sums, products and negatives of real numbers are real numbers, by Arithmetic and Order Rules in the Real Numbers, Stated for ℝ (i).

(i): Suppose $x+z=y+z$. Applying Adding a Positive Real, Cancelling an Added Real, and Regrouping Four Summands in ℝ (ii) twice, once to $x$ and $z$ and once to $y$ and $z$, $x=(x+z)-z=(y+z)-z=y.$

(ii): By The Real Numbers, with the Natural Numbers, Integers and Rationals Identified with Subsets of the Reals, and Completeness (ii), $\mathbb{R}$ with $+$, $\cdot$, $0$, $1$ and $\le$ is an ordered field. Using commutativity of multiplication from Arithmetic and Order Rules in the Real Numbers, Stated for ℝ (ii) for the first and last equalities, and the rule for a negative factor from Rules of Arithmetic and Order in an Ordered Field (ii), applied to the ordered field $\mathbb{R}$ and the elements $y$ and $x$, for the middle one, $x\cdot(-y)=(-y)\cdot x=-(y\cdot x)=-(x\cdot y).$