Multiple of Divisor Divides Multiple/Proof 2

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

Let:
 * $a \mathop \backslash b$

where $\backslash$ denotes divisibility.

Then:
 * $a c \mathop \backslash b c$

Proof
By definition, if $a \mathop \backslash b$ 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.