Intervals of Natural Numbers: Initial Segments, Adding One Element, Splitting and Shifting

In the setting of The Natural Numbers and the Natural Numbers with Zero: Arithmetic, Order, Induction and Recursion, let $m,n,p\in\mathbb{N}_{0}$, and let intervals and $[n]$ be as in Intervals of Natural Numbers §interval and Intervals of Natural Numbers §segment.

¶ $[n]=\{k\in\mathbb{N}:k\le n\}$, $[0]=\emptyset$ and $[1]=\{1\}$.

¶ $[n+1]=[n]\cup\{n+1\}$ and $n+1\notin[n]$.

¶ If $m\le n+1$, then $\{m,\dots,n+p\}=\{m,\dots,n\}\cup\{n+1,\dots,n+p\}$, and these two intervals are disjoint.

¶ The map $k\mapsto k+p$ is a bijection from $\{m,\dots,n\}$ onto $\{m+p,\dots,n+p\}$.

¶ $[m]\subseteq[n]$ if and only if $m\le n$.