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 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 real : set const SNoLev : set set const ordsucc : set set const omega : 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 SNoLe : set set prop term <= = SNoLe infix <= 2020 2020 lemma !x:set.!y:set.x iIn setexp real omega -> y iIn setexp real omega -> (!z:set.z iIn omega -> ap x z <= ap y z & ap x z <= ap x (ordsucc z) & ap y (ordsucc z) <= ap y z) -> ~ (?z:set.z iIn real & !w:set.w iIn omega -> ap x w <= z & z <= ap y w) -> (!z:set.z iIn omega -> SNo (ap x z)) -> ~ !z:set.z iIn omega -> SNo (ap y z) var x:set var y:set hyp x iIn setexp real omega hyp y iIn setexp real omega hyp !z:set.z iIn omega -> ap x z <= ap y z & ap x z <= ap x (ordsucc z) & ap y (ordsucc z) <= ap y z hyp ~ ?z:set.z iIn real & !w:set.w iIn omega -> ap x w <= z & z <= ap y w claim ~ !z:set.z iIn omega -> SNo (ap x z)