G-Tower is Nest

Theorem
Let $M$ be a class.

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

Let $M$ be a $g$-tower under $g$.

Then $M$ is a nest.

Proof
We need to show that:
 * $\forall x, y \in M: x \subseteq y$ or $y \subseteq x$

First a lemma:

Lemma
That is:

By definition of $g$-tower:


 * $M$ is minimally superinductive under $g$.

Hence by the Principle of Superinduction:
 * $\forall x, y \in M: \tuple {x, y} \in \RR$

That is:
 * $\forall x, y \in M: \map g x \subseteq y \land y \subseteq x$