Product of Quotients of Real Numbers

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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

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

$\blacksquare$


Sources