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 ordinal : set prop const ordsucc : set set const omega : set axiom ordsucc_omega_ordinal: ordinal (ordsucc omega) const SNoS_ : set set const SNoLev : set set const SNoCut : set set set axiom SNoCutP_SNoCut_omega: !x:set.Subq x (SNoS_ omega) -> !y:set.Subq y (SNoS_ omega) -> SNoCutP x y -> SNoLev (SNoCut x y) iIn ordsucc omega const SNo_ : set set prop axiom SNoLev_: !x:set.SNo x -> SNo_ (SNoLev x) x axiom SNoS_I: !x:set.ordinal x -> !y:set.!z:set.z iIn x -> SNo_ z y -> y iIn SNoS_ x const nat_p : set prop const Empty : set axiom nat_0: nat_p Empty axiom nat_p_omega: !x:set.nat_p x -> x iIn omega axiom omega_ordinal: ordinal omega 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 add_SNo : set set set term + = add_SNo infix + 2281 2280 const minus_SNo : set set term - = minus_SNo axiom SNoLt_minus_pos: !x:set.!y:set.SNo x -> SNo y -> x < y -> Empty < y + - x const real : set const abs_SNo : set set 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 Sep : set (set prop) set axiom SepE: !x:set.!p:set prop.!y:set.y iIn Sep x p -> y iIn x & p y axiom SNoS_E2: !x:set.ordinal x -> !y:set.y iIn SNoS_ x -> !P:prop.(SNoLev y iIn x -> ordinal (SNoLev y) -> SNo y -> SNo_ (SNoLev y) y -> P) -> P axiom FalseE: ~ False axiom real_I: !x:set.x iIn SNoS_ (ordsucc omega) -> x != omega -> x != - omega -> (!y:set.y iIn SNoS_ omega -> (!z:set.z iIn omega -> abs_SNo (y + - x) < eps_ z) -> y = x) -> x iIn real const SNoLe : set set prop term <= = SNoLe infix <= 2020 2020 lemma !x:set.!y:set.y iIn setexp real omega -> SNo (SNoCut (Sep (SNoS_ omega) \z:set.?w:set.w iIn omega & z < ap x w) (Sep (SNoS_ omega) \z:set.?w:set.w iIn omega & ap y w < z)) -> (!z:set.z iIn omega -> SNoCut (Sep (SNoS_ omega) \w:set.?u:set.u iIn omega & w < ap x u) (Sep (SNoS_ omega) \w:set.?u:set.u iIn omega & ap y u < w) <= ap y z) -> SNoCut (Sep (SNoS_ omega) \z:set.?w:set.w iIn omega & z < ap x w) (Sep (SNoS_ omega) \z:set.?w:set.w iIn omega & ap y w < z) = omega -> ~ SNoCut (Sep (SNoS_ omega) \z:set.?w:set.w iIn omega & z < ap x w) (Sep (SNoS_ omega) \z:set.?w:set.w iIn omega & ap y w < z) <= ap y Empty lemma !x:set.!y:set.x iIn setexp real omega -> SNo (SNoCut (Sep (SNoS_ omega) \z:set.?w:set.w iIn omega & z < ap x w) (Sep (SNoS_ omega) \z:set.?w:set.w iIn omega & ap y w < z)) -> (!z:set.z iIn omega -> ap x z <= SNoCut (Sep (SNoS_ omega) \w:set.?u:set.u iIn omega & w < ap x u) (Sep (SNoS_ omega) \w:set.?u:set.u iIn omega & ap y u < w)) -> SNoCut (Sep (SNoS_ omega) \z:set.?w:set.w iIn omega & z < ap x w) (Sep (SNoS_ omega) \z:set.?w:set.w iIn omega & ap y w < z) = - omega -> ~ ap x Empty <= SNoCut (Sep (SNoS_ omega) \z:set.?w:set.w iIn omega & z < ap x w) (Sep (SNoS_ omega) \z:set.?w:set.w iIn omega & ap y w < z) lemma !x:set.!y:set.!z:set.!w:set.(!u:set.u iIn omega -> SNo (ap x u)) -> SNo (SNoCut (Sep (SNoS_ omega) \u:set.?v:set.v iIn omega & u < ap x v) (Sep (SNoS_ omega) \u:set.?v:set.v iIn omega & ap y v < u)) -> (!u:set.u iIn omega -> ap x u <= SNoCut (Sep (SNoS_ omega) \v:set.?x2:set.x2 iIn omega & v < ap x x2) (Sep (SNoS_ omega) \v:set.?x2:set.x2 iIn omega & ap y x2 < v)) -> z iIn SNoS_ omega -> (!u:set.u iIn omega -> abs_SNo (z + - SNoCut (Sep (SNoS_ omega) \v:set.?x2:set.x2 iIn omega & v < ap x x2) (Sep (SNoS_ omega) \v:set.?x2:set.x2 iIn omega & ap y x2 < v)) < eps_ u) -> SNo z -> w iIn omega -> z < ap x w -> (!u:set.u iIn SNoS_ omega -> (!v:set.v iIn omega -> abs_SNo (u + - ap x w) < eps_ v) -> u = ap x w) -> z != ap x w lemma !x:set.!y:set.!z:set.!w:set.!u:set.(!v:set.v iIn omega -> SNo (ap x v)) -> SNo (SNoCut (Sep (SNoS_ omega) \v:set.?x2:set.x2 iIn omega & v < ap x x2) (Sep (SNoS_ omega) \v:set.?x2:set.x2 iIn omega & ap y x2 < v)) -> (!v:set.v iIn omega -> ap x v <= SNoCut (Sep (SNoS_ omega) \x2:set.?y2:set.y2 iIn omega & x2 < ap x y2) (Sep (SNoS_ omega) \x2:set.?y2:set.y2 iIn omega & ap y y2 < x2)) -> (!v:set.v iIn omega -> abs_SNo (z + - SNoCut (Sep (SNoS_ omega) \x2:set.?y2:set.y2 iIn omega & x2 < ap x y2) (Sep (SNoS_ omega) \x2:set.?y2:set.y2 iIn omega & ap y y2 < x2)) < eps_ v) -> SNo z -> w iIn omega -> z < ap x w -> u iIn omega -> Empty < ap x w + - z -> abs_SNo (z + - ap x w) < eps_ u lemma !x:set.!y:set.!z:set.!w:set.(!u:set.u iIn omega -> SNo (ap y u)) -> SNo (SNoCut (Sep (SNoS_ omega) \u:set.?v:set.v iIn omega & u < ap x v) (Sep (SNoS_ omega) \u:set.?v:set.v iIn omega & ap y v < u)) -> (!u:set.u iIn omega -> SNoCut (Sep (SNoS_ omega) \v:set.?x2:set.x2 iIn omega & v < ap x x2) (Sep (SNoS_ omega) \v:set.?x2:set.x2 iIn omega & ap y x2 < v) <= ap y u) -> z iIn SNoS_ omega -> (!u:set.u iIn omega -> abs_SNo (z + - SNoCut (Sep (SNoS_ omega) \v:set.?x2:set.x2 iIn omega & v < ap x x2) (Sep (SNoS_ omega) \v:set.?x2:set.x2 iIn omega & ap y x2 < v)) < eps_ u) -> SNo z -> w iIn omega -> ap y w < z -> (!u:set.u iIn SNoS_ omega -> (!v:set.v iIn omega -> abs_SNo (u + - ap y w) < eps_ v) -> u = ap y w) -> z != ap y w lemma !x:set.!y:set.!z:set.!w:set.!u:set.(!v:set.v iIn omega -> SNo (ap y v)) -> SNo (SNoCut (Sep (SNoS_ omega) \v:set.?x2:set.x2 iIn omega & v < ap x x2) (Sep (SNoS_ omega) \v:set.?x2:set.x2 iIn omega & ap y x2 < v)) -> (!v:set.v iIn omega -> SNoCut (Sep (SNoS_ omega) \x2:set.?y2:set.y2 iIn omega & x2 < ap x y2) (Sep (SNoS_ omega) \x2:set.?y2:set.y2 iIn omega & ap y y2 < x2) <= ap y v) -> (!v:set.v iIn omega -> abs_SNo (z + - SNoCut (Sep (SNoS_ omega) \x2:set.?y2:set.y2 iIn omega & x2 < ap x y2) (Sep (SNoS_ omega) \x2:set.?y2:set.y2 iIn omega & ap y y2 < x2)) < eps_ v) -> SNo z -> w iIn omega -> ap y w < z -> u iIn omega -> Empty < z + - ap y w -> abs_SNo (z + - ap y w) < eps_ u const SNoEq_ : set set set prop var x:set var y:set hyp x iIn setexp real omega hyp y iIn setexp real omega hyp ~ ?z:set.z iIn real & !w:set.w iIn omega -> ap x w <= z & z <= ap y w hyp !z:set.z iIn omega -> SNo (ap x z) hyp !z:set.z iIn omega -> SNo (ap y z) hyp Subq (Sep (SNoS_ omega) \z:set.?w:set.w iIn omega & z < ap x w) (SNoS_ omega) hyp Subq (Sep (SNoS_ omega) \z:set.?w:set.w iIn omega & ap y w < z) (SNoS_ omega) hyp SNoCutP (Sep (SNoS_ omega) \z:set.?w:set.w iIn omega & z < ap x w) (Sep (SNoS_ omega) \z:set.?w:set.w iIn omega & ap y w < z) hyp SNo (SNoCut (Sep (SNoS_ omega) \z:set.?w:set.w iIn omega & z < ap x w) (Sep (SNoS_ omega) \z:set.?w:set.w iIn omega & ap y w < z)) hyp !z:set.SNo z -> (!w:set.w iIn Sep (SNoS_ omega) (\u:set.?v:set.v iIn omega & u < ap x v) -> w < z) -> (!w:set.w iIn Sep (SNoS_ omega) (\u:set.?v:set.v iIn omega & ap y v < u) -> z < w) -> Subq (SNoLev (SNoCut (Sep (SNoS_ omega) \w:set.?u:set.u iIn omega & w < ap x u) (Sep (SNoS_ omega) \w:set.?u:set.u iIn omega & ap y u < w))) (SNoLev z) & SNoEq_ (SNoLev (SNoCut (Sep (SNoS_ omega) \w:set.?u:set.u iIn omega & w < ap x u) (Sep (SNoS_ omega) \w:set.?u:set.u iIn omega & ap y u < w))) (SNoCut (Sep (SNoS_ omega) \w:set.?u:set.u iIn omega & w < ap x u) (Sep (SNoS_ omega) \w:set.?u:set.u iIn omega & ap y u < w)) z hyp !z:set.z iIn omega -> ap x z <= SNoCut (Sep (SNoS_ omega) \w:set.?u:set.u iIn omega & w < ap x u) (Sep (SNoS_ omega) \w:set.?u:set.u iIn omega & ap y u < w) hyp !z:set.z iIn omega -> SNoCut (Sep (SNoS_ omega) \w:set.?u:set.u iIn omega & w < ap x u) (Sep (SNoS_ omega) \w:set.?u:set.u iIn omega & ap y u < w) <= ap y z hyp !z:set.z iIn omega -> ap x z <= SNoCut (Sep (SNoS_ omega) \w:set.?u:set.u iIn omega & w < ap x u) (Sep (SNoS_ omega) \w:set.?u:set.u iIn omega & ap y u < w) & SNoCut (Sep (SNoS_ omega) \w:set.?u:set.u iIn omega & w < ap x u) (Sep (SNoS_ omega) \w:set.?u:set.u iIn omega & ap y u < w) <= ap y z claim ~ !z:set.z iIn SNoS_ omega -> z iIn Sep (SNoS_ omega) (\w:set.?u:set.u iIn omega & w < ap x u) | z iIn Sep (SNoS_ omega) \w:set.?u:set.u iIn omega & ap y u < w