# Product of Quotients of Real Numbers

## Theorem

$\forall x, w \in \R, y, z \in \R_{\ne 0}: \dfrac x y \times \dfrac w z = \dfrac {x \times w} {y \times z}$

## Proof

 $\displaystyle \frac x y \times \frac w z$ $=$ $\displaystyle x \times \frac 1 y \times w \times \frac 1 z$ Definition of Real Division $\displaystyle$ $=$ $\displaystyle x \times w \times \frac 1 y \times \frac 1 z$ Real Number Axioms: $\R M 2$: Commutativity $\displaystyle$ $=$ $\displaystyle \paren {x \times w} \times \paren {\frac 1 y \times \frac 1 z}$ Real Number Axioms: $\R M 1$: Associativity $\displaystyle$ $=$ $\displaystyle \paren {x \times w} \times \frac 1 {y \times z}$ Product of Reciprocals of Real Numbers $\displaystyle$ $=$ $\displaystyle \dfrac {x \times w} {y \times z}$ Definition of Real Division

$\blacksquare$