Product of Real Number with Quotient

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Theorem

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


Proof

\(\displaystyle \frac {a \times x} y\) \(=\) \(\displaystyle \paren {a \times x} \times \frac 1 y\) Definition of Real Division
\(\displaystyle \) \(=\) \(\displaystyle a \times \paren {x \times \frac 1 y}\) Real Number Axioms: $\R M1$: Associativity
\(\displaystyle \) \(=\) \(\displaystyle a \times \frac x y\) Definition of Real Division

$\blacksquare$


Sources