The Quotient 1/x, Order Rules for Sums, and Nesting of Closed Intervals 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 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), and $<$ is the strict relation of $\le$. Let $x$, $y$, $z$, $p$, $a$, $b$, $c$ and $d$ be real numbers.

(i) If $x\neq0$, then the quotient $1/x$ equals the reciprocal $x^{-1}$.

(ii) If $1\le p$, then $0<p$.

(iii) If $0\le x$ and $0\le y$, then $0\le x+y$.

(iv) If $0\le y$, then $x\le x+y$.

(v) If $x\le y$, then $z+x\le z+y$.

(vi) If $a\le c$ and $d\le b$, then the closed intervals satisfy $[c,d]\subseteq[a,b]$.