Relation on Slowly Progressing Mapping which fulfils conditions of General Double Induction Principle

Theorem
Let $M$ be a class.

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

Let $\map \RR {x, y}$ be the relation on $M$ defined as:
 * $\forall x, y \in M: \tuple {x, y} \in \R \iff x \subseteq y \lor y \subseteq x$

Then $\RR$ satisfies the conditions of the General Double Induction Principle:


 * $({\text D'}_1): \quad \map \RR {x, 0}$ and $\map \RR {0, x}$ hold for every $x \in M$


 * $({\text D'}_2): \quad \forall x, y \in M: \paren {\map \RR {x, y} \land \map \RR {x, \map g y} \land \map \RR {\map g x, y} } \implies \map \RR {\map g x, \map g y}$