const Sep : set (set prop) set const SNoS_ : set set const ordsucc : set set const omega : set const minus_SNo : set set term - = minus_SNo const In : set set prop term iIn = In infix iIn 2000 2000 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 term real = Sep (SNoS_ (ordsucc omega)) \x:set.x != omega & x != - omega & !y:set.y iIn SNoS_ omega -> (!z:set.z iIn omega -> abs_SNo (y + - x) < eps_ z) -> y = x const ordinal : set prop axiom ordsucc_omega_ordinal: ordinal (ordsucc omega) axiom omega_ordinal: ordinal omega const SNo : set prop const SNoLev : set set const SNoLe : set set prop term <= = SNoLe infix <= 2020 2020 axiom ordinal_SNoLev_max_2: !x:set.ordinal x -> !y:set.SNo y -> SNoLev y iIn ordsucc x -> y <= x 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 axiom SepE: !x:set.!p:set prop.!y:set.y iIn Sep x p -> y iIn x & p y lemma !x:set.!P:prop.(SNo x -> SNoLev x iIn ordsucc omega -> x iIn SNoS_ (ordsucc omega) -> - omega < x -> x < omega -> (!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)) -> P) -> x iIn SNoS_ (ordsucc omega) -> x != omega -> x != - omega -> (!y:set.y iIn SNoS_ omega -> (!z:set.z iIn omega -> abs_SNo (y + - x) < eps_ z) -> y = x) -> SNoLev x iIn ordsucc omega -> SNo x -> x < omega -> P lemma !x:set.x != omega -> SNoLev x iIn ordsucc omega -> SNo x -> x <= omega -> x < omega claim !x:set.x iIn real -> !P:prop.(SNo x -> SNoLev x iIn ordsucc omega -> x iIn SNoS_ (ordsucc omega) -> - omega < x -> x < omega -> (!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)) -> P) -> P