Remark. Since every well-ordered set is well-founded, the well-founded recursion theorem is a generalization of the transfinite recursion theorem. Notice that the $G$ here is a function in two arguments, and that it is necessary to specify the element $x$ in the first argument (in contrast with the $G$ in the transfinite recursion theorem), since it is possible that $\operatorname{seg}(a)=\operatorname{seg}(b)$ for $a\ne b$ in a well-founded set.

By converting $G$ into a formula ($\varphi(x,y,z)$ such that for all $x,y$ , there is a unique $z$ such that $\varphi(x,y,z)$ ), then the above theorem can be proved in ZF (with the aid of the well-founded induction). The proof is similar to the proof of the transfinite recursion theorem.