Isomorphism to Closed Interval

Theorem
Let $m, n \in \N$ such that $m < n$.

Then:
 * $\left|{\left[{m + 1 \,.\,.\, n}\right]}\right| = n - m$

Let $h: \N_{n - m} \to \left[{m + 1 \,.\,.\, n}\right]$ be the mapping defined as:


 * $\forall x \in \N_{n - m}: h \left({x}\right) = x + m + 1$

Then $h$ is the unique order isomorphism as defined in Unique Isomorphism between Finite Totally Ordered Sets, where the orderings on $\left[{m + 1 \,.\,.\, n}\right]$ and $\N_{n - m}$ are those induced by the ordering of $\N$.

Proof
It suffices to show that $h$ is an order isomorphism.

To this end, remark that, for all $x, y \in \N_{n - m}$:

proving $h$ is an injection, and so a bijection, from Same Cardinality Bijective Injective Surjective.

By Ordering on Natural Numbers Compatible with Addition and Natural Number Addition is Cancellable for Ordering, it follows that:


 * $x \le y \iff h \left({x}\right) \le h \left({y}\right)$

so $h$ is an order isomorphism.