Sandwich Principle for G-Towers/Corollary 2

Theorem
Let $M$ be a class.

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

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

Let $x, y \in M$ be arbitrary such that $x \subseteq y$.

Then:
 * $\map g x \subseteq \map g y$

Proof
From Lemma $2$ of $g$-Tower is Nest we have that:


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

From the Sandwich Principle: Corollary 2:


 * $\forall x, y \in M: x \subseteq y\implies \map g x \subseteq \map g y$

Also see

 * Sandwich Principle