Double Induction Principle/General

Theorem
Let $M$ be a class.

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

Let $M$ be a minimally inductive class under $g$.

Let $\RR$ be a relation which satisfies the following conditions:


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


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

Then $\map \RR {x, y}$ holds for every $x, y \in M$.