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 const Empty : set 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 In : set set prop term iIn = In infix iIn 2000 2000 const SNoS_ : set set const omega : set const abs_SNo : set set const eps_ : set set const ap : set set set const SNoCut : set set set const Repl : set (set set) set lemma !x:set.!y:set.!z:set.!w:set.SNo x -> (!u:set.u iIn SNoS_ omega -> (!v:set.v iIn omega -> abs_SNo (u + - x) < eps_ v) -> u = x) -> (!u:set.u iIn omega -> ap y u < x & x < ap y u + eps_ u & !v:set.v iIn u -> ap y v < ap y u) -> (!u:set.u iIn omega -> SNo (ap y u)) -> SNo (SNoCut (Repl omega (ap y)) (Repl omega (ap z))) -> (!u:set.u iIn omega -> ap y u < SNoCut (Repl omega (ap y)) (Repl omega (ap z))) -> SNo w -> w < x -> w iIn SNoS_ omega -> Empty < x + - w -> w < SNoCut (Repl omega (ap y)) (Repl omega (ap z)) const SNoLev : set set const real : set var x:set var y:set var z:set var w:set hyp x iIn real hyp SNo x hyp !u:set.u iIn SNoS_ omega -> (!v:set.v iIn omega -> abs_SNo (u + - x) < eps_ v) -> u = x hyp !u:set.u iIn omega -> ap y u < x & x < ap y u + eps_ u & !v:set.v iIn u -> ap y v < ap y u hyp !u:set.u iIn omega -> SNo (ap y u) hyp SNo (SNoCut (Repl omega (ap y)) (Repl omega (ap z))) hyp !u:set.u iIn omega -> ap y u < SNoCut (Repl omega (ap y)) (Repl omega (ap z)) hyp SNo w hyp SNoLev w iIn SNoLev x hyp w < x claim w iIn SNoS_ omega -> w < SNoCut (Repl omega (ap y)) (Repl omega (ap z))