Transfinite Recursion Theorem/Formulation 2/Lemma

Lemma for Mapping on Ordinal Sequences has Mapping on Ordinals such that Image of Ordinal is Image of Restriction to Ordinal
Let $\On$ denote the class of all ordinals.

Let $S$ denote the class of all ordinal sequences.

Let $g$ be a mapping such that $S \subseteq \Dom g$.

There exists an extending operation $E$ such that for every mapping $F$ on $\On$:


 * $\forall \alpha \in \On: \map {\paren {\map E {F \restriction \alpha} } } \alpha = \map g {f \restriction \alpha}$