Let $X$ be a set and let $\ast$ be a \textbf{binary operation} on $X$, that is, a map $\ast:X\times X\to X$, whose value at $(x,y)$ is written $x\ast y$. Let $\mathbb{N}$ be the set of \reftext{def:natural-numbers-2026a}{natural numbers} with successor map $S$ as in that definition, let $n\in\mathbb{N}$, let $[n]$ be the \reftext{def:initial-segment-natural-numbers-2026a}{initial segment} determined by $n$, and let $a:[n]\to X$ be a map, whose value at $k$ is written $a_{k}$.

Then there is exactly one map $\sigma:[n]\to X$ such that

and

(Here $1\in[n]$ and, whenever $S(m)\in[n]$, also $m\in[n]$, by \ref{lem:initial-segment-basic-2026a} and \ref{lem:order-natural-numbers-2026a}, so both displayed conditions are meaningful.)