Dirchlet Convolution is Associative

The following is one of the properties of Dirichlet convolutions. The commutative law was proven by writing  ${\displaystyle n}$ as a product of two factors i.e. ${\displaystyle n=cd}$, and writing ${\displaystyle c}$ instead of the more cumbersome ${\displaystyle \frac{n}{d}}$.  We extend this technique to three factors here.  By a summation over ${\displaystyle n=uvw}$, I mean: take all possible combinations of ${\displaystyle u}$, ${\displaystyle v}$ and ${\displaystyle w}$ as long as ${\displaystyle n=uvw}$.


Theorem

Proof:-
For any ${\displaystyle n}$ we have
   ${\displaystyle \left[\left(f\text{*}g\right)\text{*}h\right]\left(n\right) }$
= ${\displaystyle \sum _{d|n}\left(f\text{*}g\right)\left(d\right) h\left(\frac{n}{d}\right)}$
= ${\displaystyle \sum _{n=dw}\left(f\text{*}g\right)\left(d\right) h\left(w\right)}$
= ${\displaystyle \sum _{n=dw}\{\sum _{d=uv}f\left(u\right) g\left(v\right)\} h\left(w\right)}$
= ${\displaystyle \sum _{n=uvw}f\left(u\right) g\left(v\right) h\left(w\right)}$
= ${\displaystyle \sum _{n=uc}f\left(u\right) \{\sum _{c=vw}g\left(v\right) h\left(w\right)\}}$
= ${\displaystyle \sum _{n=uc}f\left(u\right) \left(g\text{*}h\right)\left(c\right)}$
= ${\displaystyle \left[f\text{*}\left(g\text{*}h\right)\right]\left(n\right) }$
(proven)