# Reciprocal of Quotient of Real Numbers

## Theorem

$\forall x, y \in \R_{\ne 0}: \dfrac 1 {x / y} = \dfrac y x$

## Proof

 $\displaystyle \dfrac 1 {x / y}$ $=$ $\displaystyle \frac 1 {x \times \dfrac 1 y}$ Definition of Real Division $\displaystyle$ $=$ $\displaystyle 1 \times \frac 1 {x \times \dfrac 1 y}$ Real Number Axioms: $\R \text M 3$: Identity $\displaystyle$ $=$ $\displaystyle \paren {y \times \frac 1 y} \times \frac 1 {x \times \dfrac 1 y}$ Real Number Axioms: $\R \text M 4$: Inverses $\displaystyle$ $=$ $\displaystyle y \times \paren {\frac 1 y \times \frac 1 {x \times \dfrac 1 y} }$ Real Number Axioms: $\R \text M 1$: Associativity $\displaystyle$ $=$ $\displaystyle y \times \frac 1 {y \times \paren {x \times \dfrac 1 y} }$ Product of Reciprocals of Real Numbers $\displaystyle$ $=$ $\displaystyle y \times \frac 1 {x \times \paren {y \times \dfrac 1 y} }$ Real Number Axioms: $\R \text M 1$: Associativity and $\R \text M 2$: Commutativity $\displaystyle$ $=$ $\displaystyle y \times \frac 1 {x \times 1}$ Real Number Axioms: $\R \text M 4$: Inverses $\displaystyle$ $=$ $\displaystyle y \times \frac 1 x$ Real Number Axioms: $\R \text M 3$: Identity $\displaystyle$ $=$ $\displaystyle \frac y x$ Definition of Real Division

$\blacksquare$