const ordinal : set prop const omega : set axiom omega_ordinal: ordinal omega const In : set set prop term iIn = In infix iIn 2000 2000 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 SNoS_ : set set const SNoLev : set set const SNo_ : set set prop axiom SNoS_E2: !x:set.ordinal x -> !y:set.y iIn SNoS_ x -> !P:prop.(SNoLev y iIn x -> ordinal (SNoLev y) -> SNo y -> SNo_ (SNoLev y) y -> P) -> P 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.x iIn SNoS_ omega -> SNo x -> y iIn omega -> SNo (eps_ (ordsucc y)) -> ?z:set.z iIn SNoS_ omega & (z < x & x < z + eps_ y) claim !x:set.x iIn SNoS_ omega -> !y:set.y iIn omega -> ?z:set.z iIn SNoS_ omega & (z < x & x < z + eps_ y)