Minimally Closed Class under Progressing Mapping induces Nest/Proof

Proof
The fact that $g$ is a progressing mapping and that $N$ is closed under $g$ gives us that:


 * $\forall x \in N: x \subseteq \map f x \in N$

Let $\RR$ be the relation on $M$ defined as:


 * $\forall x, y \in M: \map \RR {x, y} \iff \map g x \subseteq y \lor y \subseteq x$

We are given that $g$ is a progressing mapping.

From the Progressing Function Lemma, we have that:

From the Double Induction Principle for Minimally Closed Class, if $\RR$ is a relation on $M$ which satisfies:

then $\map \RR {x, y}$ holds for all $x, y \in M$.

Thus:
 * $\forall x, y \in M: \map g x \subseteq y \lor y \subseteq x$

follows directly.

Then the fact that:
 * $\map g x \subseteq y \lor y \subseteq x$

implies that:
 * $x \subseteq y \lor y \subseteq x$

because:
 * $x \subseteq \map g x$

Hence the result by definition of a nest.