Product of Quotients of Real Numbers

Theorem

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

$\ds \frac x y \times \frac w z$

$=$

$\ds x \times \frac 1 y \times w \times \frac 1 z$

Definition of Real Division

$\ds $

$=$

$\ds x \times w \times \frac 1 y \times \frac 1 z$

Real Number Axiom $\R \text M2$: Commutativity of Multiplication

$\ds $

$=$

$\ds \paren {x \times w} \times \paren {\frac 1 y \times \frac 1 z}$

Real Number Axiom $\R \text M1$: Associativity of Multiplication

$\ds $

$=$

$\ds \paren {x \times w} \times \frac 1 {y \times z}$

$\ds $

$=$

$\ds \dfrac {x \times w} {y \times z}$

Definition of Real Division

$\blacksquare$

2000: James R. Munkres: Topology (2nd ed.) ... (previous) ... (next): $1$: Set Theory and Logic: $\S 4$: The Integers and the Real Numbers: Exercise $1 \ \text{(n)}$