This proof is very similar to the proof of the transfinite induction theorem. Suppose $\Phi$ is defined for a well-founded set $(S,R)$ , and suppose $\Phi$ is not true for every $a\in S$ . Assume further that $\Phi$ satisfies requirements 1 and 2 of the statement. Since $R$ is a well-founded relation, the set $\{a\in S:\neg\Phi(a)\}$ has an $R$ minimal element $r$ . This element is either an $R$ minimal element of $S$ itself, in which case condition 1 is violated, or it has $R$ predessors. In this case, we have by minimality $\Phi(s)$ for every $s$ such that $sRr$ , and by condition 2, $\Phi(r)$ is true, contradiction.