Associativity_on_Indexing_Set

Let $\left({S, \circ}\right)$ be a commutative semigroup. Let $\left \langle {x_\alpha} \right \rangle_{\alpha \mathop \in A}$ be a family of terms of $S$ indexed by a finite non-empty set $A$. Let $\left \langle {B_k} \right \rangle_{1 \mathop \le k \mathop \le n}$ be a family of distinct subsets of $A$ forming a partition of $A$. Then $\displaystyle \prod_{k \mathop = 1}^n \left({\prod_{a \mathop \in B_k} x_\alpha}\right) = \prod_{\alpha \mathop \in A} x_\alpha$