Let $f(i),i\in \mathbb N\, $ be a sequence of real or complex numbers then for natural numbers $m,n$ and $r$ holds sum transformation
$\sum_{i=0}^{mn+r}f(i)=\sum_{i=0}^{r}f(mn+i)+\sum_{i=0}^{m-1}\sum_{j=0}^{n-1}f(mj+i).$
This identity can be proved by induction by $r$. I am looking for an alternative proof.