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 -> SNo (ap z u)) -> (!u:set.u iIn omega -> (ap z u + - eps_ u) < x) -> SNo (SNoCut (Repl omega (ap y)) (Repl omega (ap z))) -> (!u:set.u iIn omega -> SNoCut (Repl omega (ap y)) (Repl omega (ap z)) < ap z u) -> SNo w -> x < w -> w iIn SNoS_ omega -> Empty < w + - x -> SNoCut (Repl omega (ap y)) (Repl omega (ap z)) < w 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 -> SNo (ap z u) hyp !u:set.u iIn omega -> (ap z u + - eps_ u) < x hyp SNo (SNoCut (Repl omega (ap y)) (Repl omega (ap z))) hyp !u:set.u iIn omega -> SNoCut (Repl omega (ap y)) (Repl omega (ap z)) < ap z u hyp SNo w hyp SNoLev w iIn SNoLev x hyp x < w claim w iIn SNoS_ omega -> SNoCut (Repl omega (ap y)) (Repl omega (ap z)) < w