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 nat_p : set prop axiom omega_nat_p: !x:set.x iIn omega -> nat_p x const ordinal : set prop axiom nat_p_ordinal: !x:set.nat_p x -> ordinal x const SNo : set prop const eps_ : set set axiom SNo_eps_: !x:set.x iIn omega -> SNo (eps_ x) const mul_SNo : set set set term * = mul_SNo infix * 2291 2290 axiom SNo_mul_SNo: !x:set.!y:set.SNo x -> SNo y -> SNo (x * y) const ordsucc : set set const Empty : set axiom SNo_1: SNo (ordsucc Empty) const SNoLt : set set prop term < = SNoLt infix < 2020 2020 axiom SNo_eps_decr: !x:set.x iIn omega -> !y:set.y iIn x -> eps_ x < eps_ y axiom pos_mul_SNo_Lt': !x:set.!y:set.!z:set.SNo x -> SNo y -> SNo z -> Empty < z -> x < y -> x * z < y * z axiom SNoLt_tra: !x:set.!y:set.!z:set.SNo x -> SNo y -> SNo z -> x < y -> y < z -> x < z axiom ordinal_trichotomy_or_impred: !x:set.!y:set.ordinal x -> ordinal y -> !P:prop.(x iIn y -> P) -> (x = y -> P) -> (y iIn x -> P) -> P const SNoS_ : set set axiom SNoS_ordsucc_omega_bdd_eps_pos: !x:set.x iIn SNoS_ (ordsucc omega) -> Empty < x -> x < omega -> ?y:set.y iIn omega & eps_ y * x < ordsucc Empty const real : 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 lemma !x:set.!y:set.!z:set.!w:set.Empty < x -> Empty < y -> ~ x * y iIn real -> SNo x -> x iIn SNoS_ (ordsucc omega) -> x < omega -> SNo y -> y iIn SNoS_ (ordsucc omega) -> y < omega -> (!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) -> SNo - x * y -> (!u:set.u iIn SNoL x -> !P:prop.(!v:set.v iIn omega -> eps_ v <= x + - u -> P) -> P) -> (!u:set.u iIn SNoR x -> !P:prop.(!v:set.v iIn omega -> eps_ v <= u + - x -> P) -> P) -> (!u:set.u iIn SNoL y -> !P:prop.(!v:set.v iIn omega -> eps_ v <= y + - u -> P) -> P) -> (!u:set.u iIn SNoR y -> !P:prop.(!v:set.v iIn omega -> eps_ v <= u + - y -> P) -> P) -> SNoCutP z w -> (!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) -> (!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) -> x * y = SNoCut z w -> (!u:set.u iIn SNoS_ omega -> (!v:set.v iIn omega -> abs_SNo (u + - x * y) < eps_ v) -> u = x * y) -> ~ ?u:set.u iIn omega & (eps_ u * x < ordsucc Empty & eps_ u * y < ordsucc Empty) const SNoLev : set set 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.SNo u -> SNoLev u iIn omega -> SNoLev u iIn SNoLev (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 claim ~ !u:set.u iIn SNoS_ omega -> (!v:set.v iIn omega -> abs_SNo (u + - x * y) < eps_ v) -> u = x * y