Fixed Point of Progressing Mapping on Minimally Closed Class is Greatest Element/Proof

Proof
Suppose $g$ has a fixed point.

Let $x$ be an element of $M$ such that $\map g x = x$.

Suppose that $y \subseteq x$.

Then by Image of Proper Subset under Progressing Mapping on Minimally Closed Class:
 * $\map g y \subseteq \map g x$

But we have that $\map g x = x$.

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

That is:
 * $\O \subseteq x$

and:
 * $y \subseteq x \implies \map g y \subseteq x$

and the result follows by the Principle of General Induction.