Adding a Positive Real, Cancelling an Added Real, and Regrouping Four Summands in ℝ

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 operation $+$ and its element $0$ are as in The Real Numbers, with the Natural Numbers, Integers and Rationals Identified with Subsets of the Reals, and Completeness (ii); $<$ is the strict relation associated with $\le$, and for $x,y\in\mathbb{R}$, $x-y$ is the difference of $x$ and $y$. Let $x$, $y$, $a$, $b$, $c$ and $d$ be real numbers.

(i) If $0<y$, then $x<x+y$.

(ii) $(x+y)-y=x$.

(iii) $(a+b)+(c+d)=(a+c)+(b+d)$.