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 nIn = \x:set.\y:set.~ x 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 omega : set const SNo : set prop const eps_ : set set axiom SNo_eps_: !x:set.x iIn omega -> SNo (eps_ x) const SNoLt : set set prop term < = SNoLt infix < 2020 2020 const Empty : set const mul_SNo : set set set term * = mul_SNo infix * 2291 2290 const real : set const SNoS_ : set set const add_SNo : set set set term + = add_SNo infix + 2281 2280 const minus_SNo : set set term - = minus_SNo const SNoL : set set const SNoLe : set set prop term <= = SNoLe infix <= 2020 2020 const SNoR : set set const SNoCutP : set set prop const SNoCut : set set set const abs_SNo : set set const ordsucc : set set lemma !x:set.!y:set.!z:set.!w:set.!u:set.Empty < x -> Empty < y -> ~ x * y iIn real -> SNo x -> SNo y -> (!v:set.v iIn omega -> !P:prop.(!x2:set.x2 iIn SNoS_ omega -> Empty < x2 -> x2 < x -> x < x2 + eps_ v -> P) -> P) -> (!v:set.v iIn omega -> !P:prop.(!x2:set.x2 iIn SNoS_ omega -> Empty < x2 -> x2 < y -> y < x2 + eps_ v -> P) -> P) -> SNo (x * y) -> SNo - x * y -> (!v:set.v iIn SNoL x -> !P:prop.(!x2:set.x2 iIn omega -> eps_ x2 <= x + - v -> P) -> P) -> (!v:set.v iIn SNoR x -> !P:prop.(!x2:set.x2 iIn omega -> eps_ x2 <= v + - x -> P) -> P) -> (!v:set.v iIn SNoL y -> !P:prop.(!x2:set.x2 iIn omega -> eps_ x2 <= y + - v -> P) -> P) -> (!v:set.v iIn SNoR y -> !P:prop.(!x2:set.x2 iIn omega -> eps_ x2 <= v + - y -> P) -> P) -> SNoCutP z w -> (!v:set.v iIn z -> !P:prop.(!x2:set.x2 iIn SNoL x -> !y2:set.y2 iIn SNoL y -> v = x2 * y + x * y2 + - x2 * y2 -> P) -> (!x2:set.x2 iIn SNoR x -> !y2:set.y2 iIn SNoR y -> v = x2 * y + x * y2 + - x2 * y2 -> P) -> P) -> (!v:set.v iIn w -> !P:prop.(!x2:set.x2 iIn SNoL x -> !y2:set.y2 iIn SNoR y -> v = x2 * y + x * y2 + - x2 * y2 -> P) -> (!x2:set.x2 iIn SNoR x -> !y2:set.y2 iIn SNoL y -> v = x2 * y + x * y2 + - x2 * y2 -> P) -> P) -> x * y = SNoCut z w -> (!v:set.v iIn SNoS_ omega -> (!x2:set.x2 iIn omega -> abs_SNo (v + - x * y) < eps_ x2) -> v = x * y) -> u iIn omega -> eps_ u * x < ordsucc Empty -> eps_ u * y < ordsucc Empty -> ~ !v:set.v iIn omega -> ?x2:set.x2 iIn SNoS_ omega & (x2 < x * y & x * y < x2 + eps_ v) lemma !x:set.!y:set.!z:set.!w:set.Empty < x -> Empty < y -> SNo x -> SNo y -> (!u:set.u iIn omega -> !P:prop.(!v:set.v iIn SNoS_ omega -> Empty < v -> v < x -> x < v + eps_ u -> P) -> P) -> (!u:set.u iIn omega -> !P:prop.(!v:set.v iIn SNoS_ omega -> Empty < v -> v < y -> y < v + eps_ u -> P) -> P) -> SNo (x * y) -> z iIn omega -> eps_ z * x < ordsucc Empty -> eps_ z * y < ordsucc Empty -> w iIn omega -> SNo (eps_ w) -> ?u:set.u iIn SNoS_ omega & (u < x * y & x * y < u + eps_ w) var x:set var y:set var z:set var w:set hyp Empty < x hyp Empty < y hyp ~ x * y iIn real hyp SNo x hyp x iIn SNoS_ (ordsucc omega) hyp x < omega hyp SNo y hyp y iIn SNoS_ (ordsucc omega) hyp y < omega hyp !u:set.u iIn omega -> !P:prop.(!v:set.v iIn SNoS_ omega -> Empty < v -> v < x -> x < v + eps_ u -> P) -> P hyp !u:set.u iIn omega -> !P:prop.(!v:set.v iIn SNoS_ omega -> Empty < v -> v < y -> y < v + eps_ u -> P) -> P hyp SNo (x * y) hyp SNo - x * y hyp !u:set.u iIn SNoL x -> !P:prop.(!v:set.v iIn omega -> eps_ v <= x + - u -> P) -> P hyp !u:set.u iIn SNoR x -> !P:prop.(!v:set.v iIn omega -> eps_ v <= u + - x -> P) -> P hyp !u:set.u iIn SNoL y -> !P:prop.(!v:set.v iIn omega -> eps_ v <= y + - u -> P) -> P hyp !u:set.u iIn SNoR y -> !P:prop.(!v:set.v iIn omega -> eps_ v <= u + - y -> P) -> P hyp SNoCutP z w hyp !u:set.u iIn z -> !P:prop.(!v:set.v iIn SNoL x -> !x2:set.x2 iIn SNoL y -> u = v * y + x * x2 + - v * x2 -> P) -> (!v:set.v iIn SNoR x -> !x2:set.x2 iIn SNoR y -> u = v * y + x * x2 + - v * x2 -> P) -> P hyp !u:set.u iIn w -> !P:prop.(!v:set.v iIn SNoL x -> !x2:set.x2 iIn SNoR y -> u = v * y + x * x2 + - v * x2 -> P) -> (!v:set.v iIn SNoR x -> !x2:set.x2 iIn SNoL y -> u = v * y + x * x2 + - v * x2 -> P) -> P hyp x * y = SNoCut z w hyp !u:set.u iIn SNoS_ omega -> (!v:set.v iIn omega -> abs_SNo (u + - x * y) < eps_ v) -> u = x * y claim ~ ?u:set.u iIn omega & (eps_ u * x < ordsucc Empty & eps_ u * y < ordsucc Empty)