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 SNo : set prop const eps_ : set set axiom SNo_eps_: !x:set.x iIn omega -> SNo (eps_ x) const add_SNo : set set set term + = add_SNo infix + 2281 2280 axiom SNo_add_SNo: !x:set.!y:set.SNo x -> SNo y -> SNo (x + y) 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 minus_SNo : set set term - = minus_SNo lemma !x:set.!y:set.!z:set.SNo x -> y iIn omega -> z iIn SNoS_ omega -> z < x -> x < z + eps_ y -> SNo z -> SNo (z + eps_ y) -> ?w:set.w iIn SNoS_ omega & (w < - x & - x < w + eps_ y) claim !x:set.SNo x -> (!y:set.y iIn omega -> ?z:set.z iIn SNoS_ omega & (z < x & x < z + eps_ y)) -> !y:set.y iIn omega -> ?z:set.z iIn SNoS_ omega & (z < - x & - x < z + eps_ y)