Multiple of Divisor in Integral Domain Divides Multiple

Theorem
Let $a, b, c \in \Z$.

Let $a$ be a divisor of $b$, that is, $a \backslash b$.

Then $a c \backslash b c$.

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

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

Hence the result.