Dirichlet convolution is commutative

The Dirichlet convolutions was introduced here.  The following illustrates a proof technique that avoids writing ${\displaystyle \frac{n}{d}}$ over and over again.
Theorem

Proof:-
For any ${\displaystyle n}$ we have
   ${\displaystyle \left(f\text{*}g\right)\left(n\right) }$
= ${\displaystyle \sum _{n=cd}f\left(c\right)\cdot g\left(d\right)}$
= ${\displaystyle \sum _{n=cd}f\left(d\right)\cdot g\left(c\right)}$  [ as ${\displaystyle d}$ runs through all divisors of  ${\displaystyle n}$, so does ${\displaystyle c=\frac{n}{d}}$ ]
= ${\displaystyle \sum _{n=cd}g\left(c\right)\cdot f\left(d\right)}$
= ${\displaystyle \left(g\text{*}f\right)\left(n\right) }$
(proven) 

The proof technique can be extended to prove the associativity law.