Iterated Operations over an Interval: Extracting One Term and Reversing the Order

In the setting of The Natural Numbers and the Natural Numbers with Zero: Arithmetic, Order, Induction and Recursion, let $\ast$ be an associative and commutative binary operation on a set $X$, let $[n]$ be as in Intervals of Natural Numbers §segment, and let iterated operations over intervals be as in Sums and Products over a Finite Set and over an Interval §intervals and Sums and Products over a Finite Set and over an Interval §empty.

¶ Let $n\in\mathbb{N}_{0}$, with $n\ge1$ if $\ast$ has no neutral element, let $j\in[n+1]$ with $g_{j}$ as in Bijections between Intervals: the Gap Map and the Reversal §gap, and let $a:[n+1]\to X$. Then

$$\mathop{\ast}\limits_{k=1}^{n+1}a_{k}=\Big(\mathop{\ast}\limits_{k=1}^{n}a_{g_{j}(k)}\Big)\ast a_{j}.$$

¶ Let $N\in\mathbb{N}$, with $N+1-k$ for $k\in[N]$ as in Bijections between Intervals: the Gap Map and the Reversal §reversal, and let $a:[N]\to X$. Then

$$\mathop{\ast}\limits_{k=1}^{N}a_{N+1-k}=\mathop{\ast}\limits_{k=1}^{N}a_{k}.$$