const In : set set prop term iIn = In infix iIn 2000 2000 const omega : set const ordsucc : set set axiom omega_ordsucc: !x:set.x iIn omega -> ordsucc x iIn omega const SNoLt : set set prop term < = SNoLt infix < 2020 2020 const Empty : set const eps_ : set set axiom SNo_eps_pos: !x:set.x iIn omega -> Empty < eps_ x const nat_p : set prop const add_SNo : set set set term + = add_SNo infix + 2281 2280 const SNo : set prop const SNoCut : set set set const Sing : set set const Repl : set (set set) set lemma !x:set.nat_p x -> (!y:set.y iIn x -> eps_ (ordsucc y) + eps_ (ordsucc y) = eps_ y) -> x iIn omega -> SNo (eps_ (ordsucc x)) -> Empty < eps_ (ordsucc x) -> eps_ (ordsucc x) + eps_ (ordsucc x) = SNoCut (Sing Empty) (Repl x eps_) var x:set hyp nat_p x hyp !y:set.y iIn x -> eps_ (ordsucc y) + eps_ (ordsucc y) = eps_ y hyp x iIn omega claim SNo (eps_ (ordsucc x)) -> eps_ (ordsucc x) + eps_ (ordsucc x) = SNoCut (Sing Empty) (Repl x eps_)