const Pi : set (set set) set term setexp = \x:set.\y:set.Pi y \z:set.x const ordinal : set prop const omega : set axiom omega_ordinal: ordinal omega const In : set set prop term iIn = In infix iIn 2000 2000 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 SNoS_ : set set const SNoLev : set set const SNo : set prop const SNo_ : set set prop 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 const SNoLt : set set prop term < = SNoLt infix < 2020 2020 const abs_SNo : set set const add_SNo : set set set term + = add_SNo infix + 2281 2280 const minus_SNo : set set term - = minus_SNo const eps_ : set set axiom SNo_prereal_decr_upper_approx: !x:set.SNo x -> (!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)) -> ?y:set.y iIn setexp (SNoS_ omega) omega & !z:set.z iIn omega -> (ap y z + - eps_ z) < x & x < ap y z & !w:set.w iIn z -> ap y z < ap y w axiom SNo_prereal_incr_lower_approx: !x:set.SNo x -> (!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)) -> ?y:set.y iIn setexp (SNoS_ omega) omega & !z:set.z iIn omega -> ap y z < x & x < ap y z + eps_ z & !w:set.w iIn z -> ap y w < ap y z const real : set const ordsucc : 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 SNoCutP : set set prop const Repl : set (set set) set const SNoCut : set set set lemma !x:set.!P:prop.!y:set.!z:set.x iIn real -> (!w:set.w iIn setexp (SNoS_ omega) omega -> !u:set.u iIn setexp (SNoS_ omega) omega -> (!v:set.v iIn omega -> ap w v < x) -> (!v:set.v iIn omega -> x < ap w v + eps_ v) -> (!v:set.v iIn omega -> !x2:set.x2 iIn v -> ap w x2 < ap w v) -> (!v:set.v iIn omega -> (ap u v + - eps_ v) < x) -> (!v:set.v iIn omega -> x < ap u v) -> (!v:set.v iIn omega -> !x2:set.x2 iIn v -> ap u v < ap u x2) -> SNoCutP (Repl omega (ap w)) (Repl omega (ap u)) -> x = SNoCut (Repl omega (ap w)) (Repl omega (ap u)) -> P) -> SNo x -> (!w:set.w iIn SNoS_ omega -> (!u:set.u iIn omega -> abs_SNo (w + - x) < eps_ u) -> w = x) -> y iIn setexp (SNoS_ omega) omega -> (!w:set.w iIn omega -> ap y w < x & x < ap y w + eps_ w & !u:set.u iIn w -> ap y u < ap y w) -> z iIn setexp (SNoS_ omega) omega -> (!w:set.w iIn omega -> (ap z w + - eps_ w) < x & x < ap z w & !u:set.u iIn w -> ap z w < ap z u) -> (!w:set.w iIn omega -> SNo (ap y w)) -> P claim !x:set.x iIn real -> !P:prop.(!y:set.y iIn setexp (SNoS_ omega) omega -> !z:set.z iIn setexp (SNoS_ omega) omega -> (!w:set.w iIn omega -> ap y w < x) -> (!w:set.w iIn omega -> x < ap y w + eps_ w) -> (!w:set.w iIn omega -> !u:set.u iIn w -> ap y u < ap y w) -> (!w:set.w iIn omega -> (ap z w + - eps_ w) < x) -> (!w:set.w iIn omega -> x < ap z w) -> (!w:set.w iIn omega -> !u:set.u iIn w -> ap z w < ap z u) -> SNoCutP (Repl omega (ap y)) (Repl omega (ap z)) -> x = SNoCut (Repl omega (ap y)) (Repl omega (ap z)) -> P) -> P