Let $A$ be a ring and $f,g,h\in A[X]$. I want to show that $(fg)h=f(gh)$, where $(xy)_n:=\sum_{n=j+k}x_jy_k$ for all $n\in\mathbb{N}$.
Attempt: Let $n\in\mathbb{N}$. I have to show that $((fg)h)_n=(f(gh))_n$. By definition $$((fg)h)_n=\sum_{n=j+k}(fg)_jh_k=\sum_{n=j+k}\left(\sum_{j=s+t}f_sg_t\right)h_k=\sum_{n=j+k}\sum_{j=s+t}(f_sg_t)h_k.$$ Presumably that last expression is equal to $\sum_{n=(s+t)+k}(f_sg_t)h_k$. But why? What is the formal justification?