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.

Then $M$ is a nest.

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

First some lemmata:

Lemma 1
That is:

Lemma 2
But as $g: M \to M$ is a progressing mapping:


 * $\forall x \in M: x \subseteq \map g x$

The result follows by Subset Relation is Transitive.