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 SNo : set prop const eps_ : set set axiom SNo_eps_: !x:set.x iIn omega -> SNo (eps_ x) const nat_p : set prop const SNoLt : set set prop term < = SNoLt infix < 2020 2020 const add_SNo : set set set term + = add_SNo infix + 2281 2280 lemma !x:set.!y:set.nat_p x -> x iIn omega -> SNo (eps_ (ordsucc x)) -> y iIn x -> y iIn omega -> SNo (eps_ (ordsucc y)) -> (eps_ (ordsucc x) + eps_ (ordsucc x)) < eps_ (ordsucc y) + eps_ (ordsucc y) var x:set var y:set hyp nat_p x hyp x iIn omega hyp SNo (eps_ (ordsucc x)) hyp y iIn x claim y iIn omega -> (eps_ (ordsucc x) + eps_ (ordsucc x)) < eps_ (ordsucc y) + eps_ (ordsucc y)