const In : set set prop term iIn = In infix iIn 2000 2000 term Subq = \x:set.\y:set.!z:set.z iIn x -> z iIn y term TransSet = \x:set.!y:set.y iIn x -> Subq y x const Pi : set (set set) set term setexp = \x:set.\y:set.Pi y \z:set.x const SNo : set prop const SNoLt : set set prop term < = SNoLt infix < 2020 2020 term SNoCutP = \x:set.\y:set.(!z:set.z iIn x -> SNo z) & (!z:set.z iIn y -> SNo z) & !z:set.z iIn x -> !w:set.w iIn y -> z < w term nIn = \x:set.\y:set.~ x iIn y const nat_p : set prop const Empty : set axiom nat_0: nat_p Empty const omega : set axiom nat_p_omega: !x:set.nat_p x -> x iIn omega const ap : set set set axiom ap_Pi: !x:set.!f:set set.!y:set.!z:set.y iIn Pi x f -> z iIn x -> ap y z iIn f z const SNoLe : set set prop term <= = SNoLe infix <= 2020 2020 axiom SNoLeLt_tra: !x:set.!y:set.!z:set.SNo x -> SNo y -> SNo z -> x <= y -> y < z -> x < z axiom SNoLt_irref: !x:set.~ x < x const real : set const SNoLev : set set const ordsucc : set set const SNoS_ : set set const minus_SNo : set set term - = minus_SNo const abs_SNo : set set const add_SNo : set set set term + = add_SNo infix + 2281 2280 const eps_ : set set axiom real_E: !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 const SNoCut : set set set const Sep : set (set prop) set var x:set var y:set hyp y iIn setexp real omega hyp SNo (SNoCut (Sep (SNoS_ omega) \z:set.?w:set.w iIn omega & z < ap x w) (Sep (SNoS_ omega) \z:set.?w:set.w iIn omega & ap y w < z)) hyp !z:set.z iIn omega -> SNoCut (Sep (SNoS_ omega) \w:set.?u:set.u iIn omega & w < ap x u) (Sep (SNoS_ omega) \w:set.?u:set.u iIn omega & ap y u < w) <= ap y z hyp SNoCut (Sep (SNoS_ omega) \z:set.?w:set.w iIn omega & z < ap x w) (Sep (SNoS_ omega) \z:set.?w:set.w iIn omega & ap y w < z) = omega claim ~ SNoCut (Sep (SNoS_ omega) \z:set.?w:set.w iIn omega & z < ap x w) (Sep (SNoS_ omega) \z:set.?w:set.w iIn omega & ap y w < z) <= ap y Empty