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