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 $N$ defined as:
 * $\map \RR {x, y} \iff \map g x \subseteq y \lor y \subseteq x$

where $\lor$ denotes disjunction (inclusive "or").

By the Progressing Function Lemma:
 * $(1): \quad \forall y \in N: \map \RR {y, \O}$
 * $(2): \quad \forall x, y \in N: \map \RR {x, y} \land \map \RR {y, x} \implies \map \RR {x, \map g y}$