Quotient of Quotients of Real Numbers

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Theorem

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


Proof

\(\displaystyle \frac {x / y} {w / z}\) \(=\) \(\displaystyle \frac x y \times \frac 1 {w / z}\) Definition of Real Division
\(\displaystyle \) \(=\) \(\displaystyle \frac x y \times \frac z w\) Reciprocal of Quotient of Real Numbers
\(\displaystyle \) \(=\) \(\displaystyle \dfrac {x \times z} {y \times w}\) Product of Quotients of Real Numbers

$\blacksquare$


Sources