G-Tower is Closed under Mapping
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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.
$\blacksquare$