The Reciprocal and the Quotient of Real-Valued Maps with a Nonvanishing Denominator Are Maps

In the setting of Routine Facts Used Without Citation: Arithmetic, Maps, Predicativity and Indices, let $E$ be a set and let $f,\chi:E\to\mathbb{R}$ be maps with $\chi(t)\neq0$ for every $t\in E$; reciprocals and quotients are as in Negatives, Differences, Reciprocals and Quotients (ii), and $1/y=1\cdot y^{-1}$ for $y\in\mathbb{R}$ with $y\neq0$.

(i) There is exactly one map $1/\chi:E\to\mathbb{R}$ with $(1/\chi)(t)=1/\chi(t)$ for every $t\in E$.

(ii) There is exactly one map $f/\chi:E\to\mathbb{R}$ with $(f/\chi)(t)=f(t)/\chi(t)$ for every $t\in E$.