Multiple of Divisor Divides Multiple/Proof 2

Proof
By definition, if $a \mathrel \backslash b$ then $\exists d \in \Z: a d = b$.

Then:
 * $\left({a d}\right) c = b c$

that is:
 * $\left({a c}\right) d = b c$

which follows because Integer Multiplication is Commutative and Integer Multiplication is Associative.

Hence the result.