G-Tower is Closed under Mapping

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 closed under $g$:
 * $\forall x: \paren {x \in M \implies \map g x \in M}$

Proof
By definition:
 * a $g$-tower is a class which is minimally superinductive under $g$
 * a class which is minimally superinductive under $g$ is superinductive under $g$
 * a superinductive class is inductive under $g$
 * an inductive class under $g$ is closed under $g$.

Hence the result.