Chapter 3.2, Problem 29E

State and prove Theorem $3.7$ for an additive group.

Theorem $3.7$: Generalized Associative Law

Let $n\ge 2$ be a positive integer, and let $a_1,a_2,\mathrm{...},a_n$ denote elements of a group $G$. For any positive integer $m$ such that $1\le m<n$, $(a_1{a}_2\mathrm{...}{a}_m)(a_{m+1}\mathrm{...}{a}_n)=a_1{a}_2\mathrm{...}{a}_n$.