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 real : set const SNoLt : set set prop term < = SNoLt infix < 2020 2020 const add_SNo : set set set term + = add_SNo infix + 2281 2280 const eps_ : set set const minus_SNo : set set term - = minus_SNo const SNoCutP : set set prop const Repl : set (set set) set const SNoCut : set set set const abs_SNo : 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)) -> (!w:set.w iIn omega -> SNo (ap z w)) -> P var x:set var P:prop var y:set var z:set hyp x iIn real hyp !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 hyp SNo x hyp !w:set.w iIn SNoS_ omega -> (!u:set.u iIn omega -> abs_SNo (w + - x) < eps_ u) -> w = x hyp y iIn setexp (SNoS_ omega) omega hyp !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 hyp z iIn setexp (SNoS_ omega) omega hyp !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 claim (!w:set.w iIn omega -> SNo (ap y w)) -> P