const nat_p : set prop const In : set set prop term iIn = In infix iIn 2000 2000 const omega : set axiom nat_p_omega: !x:set.nat_p x -> x iIn omega axiom nat_complete_ind: !p:set prop.(!x:set.nat_p x -> (!y:set.y iIn x -> p y) -> p x) -> !x:set.nat_p x -> p x const add_SNo : set set set term + = add_SNo infix + 2281 2280 const eps_ : set set const ordsucc : set set lemma !x:set.nat_p x -> (!y:set.y iIn x -> eps_ (ordsucc y) + eps_ (ordsucc y) = eps_ y) -> x iIn omega -> eps_ (ordsucc x) + eps_ (ordsucc x) = eps_ x claim !x:set.nat_p x -> eps_ (ordsucc x) + eps_ (ordsucc x) = eps_ x