const Pi : set (set set) set term setexp = \x:set.\y:set.Pi y \z:set.x const SNo : set prop const minus_SNo : set set term - = minus_SNo axiom SNo_minus_SNo: !x:set.SNo x -> SNo - x const In : set set prop term iIn = In infix iIn 2000 2000 const SNoS_ : set set const omega : set const SNoLt : set set prop term < = SNoLt infix < 2020 2020 const abs_SNo : set set const add_SNo : set set set term + = add_SNo infix + 2281 2280 const eps_ : set set const ap : set set set lemma !x:set.SNo x -> (!y:set.y iIn SNoS_ omega -> (!z:set.z iIn omega -> abs_SNo (y + - x) < eps_ z) -> y = x) -> (!y:set.y iIn omega -> ?z:set.z iIn SNoS_ omega & (z < x & x < z + eps_ y)) -> SNo - x -> ?y:set.y iIn setexp (SNoS_ omega) omega & !z:set.z iIn omega -> (ap y z + - eps_ z) < x & x < ap y z & !w:set.w iIn z -> ap y z < ap y w claim !x:set.SNo x -> (!y:set.y iIn SNoS_ omega -> (!z:set.z iIn omega -> abs_SNo (y + - x) < eps_ z) -> y = x) -> (!y:set.y iIn omega -> ?z:set.z iIn SNoS_ omega & (z < x & x < z + eps_ y)) -> ?y:set.y iIn setexp (SNoS_ omega) omega & !z:set.z iIn omega -> (ap y z + - eps_ z) < x & x < ap y z & !w:set.w iIn z -> ap y z < ap y w