Preamble; the switching map $S_{p, q}$, the maps $\delta_m$ and $\epsilon_m$

We assume basic properties of the positive integers $\mathbb{N}$.
In particular we assume the Well Ordering Principle, POW:

• Let $\mathbb{A} \subset \mathbb{N}$.
  Then either $\mathbb{A} = \phi$, or $\mathbb{A}$ has a least element.

For $n \in\mathbb{N}$, denote by $\mathbb{N}_n$ the set:

$$\mathbb{N}_n = \{ x\in \mathbb{N}: 1\le x\le n\}.$$

For the proofs below we need a switching map which will place certain elements of a set into a desired position.
We have need only of the following switching map $S_{p, q}: \mathbb{N}_q\rightarrow \mathbb{N}_q$, which is defined for any $p \in \mathbb{N}_q$ and for any $q\in \mathbb{N}$ and which maps $\mathbb{N}_q$ to itself.
Then $S_{p, q}$ is given by the formula:

$$S_{p, q}(p) = q, \hspace{10pt}S_{p, q}(q) = p, \hspace{10pt} S_{p... ...rm{such that}\hspace{3pt} x\ne p \hspace{3pt}\textrm{and} \hspace{3pt} x \ne q.$$

It is clear that $S_{p, q}: \mathbb{N}_q\rightarrow \mathbb{N}_q$ is well-defined, and is its own inverse: we have $S_{p, q}\circ S_{p, q} = \textrm{id}_{\mathbb{N}_{q}}$. In particular, $S_{p, q}$ is bijective, injective and surjective.
Note that $S_{p, q}$ is defined even if $p = q$, in which case we have $S_{q, q} = \textrm{id}_{\mathbb{N}_q}$, the identity map.
It follows also that if $g$ and $f$ are maps into $\mathbb{N}_q$, such that $g = S_{p, q}\circ f$, then we have, using the associativity of composition:

$$S_{p, q} \circ g = S_{p, q} \circ (S_{p, q}\circ f) = (S_{p, q} \circ S_{p, q})\circ f = \textrm{id}_{\mathbb{N}_q}\circ f = f.$$

Also for the proofs below we need the properties:

• The composition of injections is an injection.
• The composition of surjections is a surjection.
• The composition of bijections is a bijection.

For each positive integer $m$, we can inject $\mathbb{N}_m$ into $\mathbb{N}$ and surject $\mathbb{N}$ onto $\mathbb{N}_m$ as follows (there is no unique way to do this):

• For each $m \in \mathbb{N}$, define a map $\delta_m: \mathbb{N}_m \rightarrow \mathbb{N}$ by the formula: $\delta_m(k) = k$, defined for any $k \in \mathbb{N}_m$. Then $\delta_m$ is well-defined and is an injection.
• For each $m \in \mathbb{N}$, define a map $\epsilon_m: \mathbb{N} \rightarrow \mathbb{N}_m$ by the formula $\epsilon_m(k) = k$, for any $k \in \mathbb{N}_m$ and $\epsilon_m(k) = m$, for any integer $k > m$.
  Then the map $\epsilon_m$ is well-defined and is a surjection.

Note that $\epsilon_m \circ \delta_m = \textrm{id}_{\mathbb{N}_m}$ and $\delta_m \circ \epsilon_m = \epsilon_m$, for each $m \in \mathbb{N}$.