General_Commutativity_Theorem

Let $\struct {S, \circ}$ be a semigroup. Let $\family {a_k}_{1 \mathop \le k \mathop \le n}$ be a sequence of elements of $S$. Suppose that $\forall i, j \in \closedint 1 n: a_i \circ a_j = a_j \circ a_i$ Then for every permutation $\sigma: \N_n \to \N_n$ $a_{\map \sigma 1} \circ \cdots \circ a_{\map \sigma n} = a_1 \circ \cdots \circ a_n$ where $\N_n$ is used here to denote the initial segment of $\N_{>0}$ $\N_n = \set {1, 2, \ldots, n}$