Sandwich Principle for Slowly Progressing Mapping/Corollary

Theorem
Let $M$ be a class.

Let $g: M \to M$ be a slowly progressing mapping on $M$.

Let $M$ be a minimally inductive class under $g$.

Let $N$ be a nest which is closed under $g$.

Then:
 * $\forall x, y \in N: \map g x \subseteq y \lor y \subseteq x$