Natural Numbers as Indices: ℕ ⊆ ℕ₀, 1 ∈ ℕ₀, and the Two Readings of an Operation from 1 to n

This helper adopts the setting of The Natural Numbers and the Natural Numbers with Zero: Arithmetic, Order, Induction and Recursion, with $\mathbb{N}$, $\mathbb{N}_{0}$ and $1$ as in The Natural Numbers and the Natural Numbers with Zero: Arithmetic, Order, Induction and Recursion §sets.

¶ $\mathbb{N}\subseteq\mathbb{N}_{0}$; that is, $n\in\mathbb{N}_{0}$ for every $n\in\mathbb{N}$.

¶ $1\in\mathbb{N}$ and $1\in\mathbb{N}_{0}$.

Let now $\ast$ be an associative and commutative binary operation on a set $X$, let $n\in\mathbb{N}$, let $[n]$ and the interval $\{1,\dots,n\}$ be as in Intervals of Natural Numbers §segment and Intervals of Natural Numbers §interval, and let $a:[n]\to X$, $k\mapsto a_{k}$, be a map. The set $[n]$ is finite by Finite Sets: the Pigeonhole Principle, Uniqueness of the Length, Subsets, Unions, Products, Images, Bounded Sets of Natural Numbers, Extreme Elements, Sets of Maps, Finite Unions and Finite Choice §naturals.

¶ The iterated operation $\mathop{\ast}\limits_{k=1}^{n}a_{k}$ of $a$ along $[n]$ equals the iterated operation over the finite set $[n]$, $\mathop{\ast}\limits_{k\in[n]}a_{k}$.

¶ $\mathop{\ast}\limits_{k=1}^{n}a_{k}$ in the sense of Sums and Products over a Finite Set and over an Interval §intervals equals $\mathop{\ast}\limits_{k\in[n]}a_{k}$.

¶ $\mathop{\ast}\limits_{k=1}^{n}a_{k}=\mathop{\ast}\limits_{k\in[n]}a_{k}$ whether the left side is read as in Iterated Operations: Finite Sums and Finite Products §iterated or as in Sums and Products over a Finite Set and over an Interval §intervals.