Cowen's Theorem

Theorem
Let $g$ be a progressing mapping.

Let $x$ be a set.

Let $\powerset x$ denote the power set of $x$.

Let $M_x$ denote the intersection of the $x$-special subsets of $\powerset x$ $g$.

Let $M$ be the class of all $x$ such that $x \in M_x$.

Then $M$ is minimally superinductive under $g$.

Lemma $7$
Recall:
 * From Lemma $2$:
 * $\O \in M$


 * From Lemma $7$:
 * $M$ is closed under $g$ relative to $x$


 * From Lemma $4$:
 * $M$ is closed under chain unions.

Hence by definition:
 * $M$ is superinductive under $g$.