Product of Real Number with Quotient

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

$\blacksquare$