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 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 const real : set const mul_SNo : set set set term * = mul_SNo infix * 2291 2290 axiom real_mul_SNo: !x:set.x iIn real -> !y:set.y iIn real -> x * y iIn real 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 Empty : set const Sep : set (set prop) set const SNoCut : set set set lemma !x:set.x iIn real -> Empty < x -> x < ordsucc Empty -> ~ (?y:set.y iIn real & x * y = ordsucc Empty) -> SNo x -> (!y:set.y iIn Sep (SNoS_ omega) (\z:set.x * z < ordsucc Empty) -> y < SNoCut (Sep (SNoS_ omega) \z:set.x * z < ordsucc Empty) (Sep (SNoS_ omega) \z:set.ordsucc Empty < x * z)) -> (!y:set.y iIn Sep (SNoS_ omega) (\z:set.ordsucc Empty < x * z) -> SNoCut (Sep (SNoS_ omega) \z:set.x * z < ordsucc Empty) (Sep (SNoS_ omega) \z:set.ordsucc Empty < x * z) < y) -> SNoCut (Sep (SNoS_ omega) \y:set.x * y < ordsucc Empty) (Sep (SNoS_ omega) \y:set.ordsucc Empty < x * y) iIn real -> SNo (SNoCut (Sep (SNoS_ omega) \y:set.x * y < ordsucc Empty) (Sep (SNoS_ omega) \y:set.ordsucc Empty < x * y)) -> (!y:set.y iIn omega -> ?z:set.z iIn SNoS_ omega & (z < SNoCut (Sep (SNoS_ omega) \w:set.x * w < ordsucc Empty) (Sep (SNoS_ omega) \w:set.ordsucc Empty < x * w) & SNoCut (Sep (SNoS_ omega) \w:set.x * w < ordsucc Empty) (Sep (SNoS_ omega) \w:set.ordsucc Empty < x * w) < z + eps_ y)) -> ~ x * SNoCut (Sep (SNoS_ omega) \y:set.x * y < ordsucc Empty) (Sep (SNoS_ omega) \y:set.ordsucc Empty < x * y) iIn real var x:set hyp x iIn real hyp Empty < x hyp x < ordsucc Empty hyp ~ ?y:set.y iIn real & x * y = ordsucc Empty hyp SNo x hyp !y:set.y iIn SNoS_ omega -> (!z:set.z iIn omega -> abs_SNo (y + - x) < eps_ z) -> y = x hyp SNo (SNoCut (Sep (SNoS_ omega) \y:set.x * y < ordsucc Empty) (Sep (SNoS_ omega) \y:set.ordsucc Empty < x * y)) hyp !y:set.y iIn Sep (SNoS_ omega) (\z:set.x * z < ordsucc Empty) -> y < SNoCut (Sep (SNoS_ omega) \z:set.x * z < ordsucc Empty) (Sep (SNoS_ omega) \z:set.ordsucc Empty < x * z) hyp !y:set.y iIn Sep (SNoS_ omega) (\z:set.ordsucc Empty < x * z) -> SNoCut (Sep (SNoS_ omega) \z:set.x * z < ordsucc Empty) (Sep (SNoS_ omega) \z:set.ordsucc Empty < x * z) < y hyp SNoCut (Sep (SNoS_ omega) \y:set.x * y < ordsucc Empty) (Sep (SNoS_ omega) \y:set.ordsucc Empty < x * y) iIn SNoS_ (ordsucc omega) claim ~ SNoCut (Sep (SNoS_ omega) \y:set.x * y < ordsucc Empty) (Sep (SNoS_ omega) \y:set.ordsucc Empty < x * y) iIn real