The Index Map Skipping One Element of [n]

In the setting of Routine Facts Used Without Citation: Arithmetic, Maps, Predicativity and Indices, let $n\in\mathbb{N}$ with $2\le n$ and let $i\in[n]$, with $[n]$ and $[n-1]$ as in Intervals of Natural Numbers §segment, where $n-1\in\mathbb{N}_{0}$ is the difference, which is defined as $1\le n$.

¶ $n-1\in\mathbb{N}$ and $(n-1)+1=n$.

¶ There is exactly one map $s:[n-1]\to[n]$ with $s(k)=k$ for every $k\in[n-1]$ with $k<i$ and $s(k)=k+1$ for every $k\in[n-1]$ with $i\le k$.

¶ For the map $s$ of clause map and all $k,l\in[n-1]$: $k<l$ if and only if $s(k)<s(l)$.

¶ For the map $s$ of clause map: $s(k)\in[n]\setminus\{i\}$ for every $k\in[n-1]$, and $s$, as a map $s:[n-1]\to[n]\setminus\{i\}$, is a bijection.