const Pi : set (set set) set term setexp = \x:set.\y:set.Pi y \z:set.x const In : set set prop term iIn = In infix iIn 2000 2000 const omega : set const SNo : set prop const eps_ : set set axiom SNo_eps_: !x:set.x iIn omega -> SNo (eps_ x) const add_SNo : set set set term + = add_SNo infix + 2281 2280 axiom SNo_add_SNo: !x:set.!y:set.SNo x -> SNo y -> SNo (x + y) const SNoLe : set set prop term <= = SNoLe infix <= 2020 2020 const SNoLt : set set prop term < = SNoLt infix < 2020 2020 axiom SNoLeLt_tra: !x:set.!y:set.!z:set.SNo x -> SNo y -> SNo z -> x <= y -> y < z -> x < z axiom add_SNo_Lt1_cancel: !x:set.!y:set.!z:set.SNo x -> SNo y -> SNo z -> (x + y) < z + y -> x < z axiom SNoLt_tra: !x:set.!y:set.!z:set.SNo x -> SNo y -> SNo z -> x < y -> y < z -> x < z const Empty : set const minus_SNo : set set term - = minus_SNo const SNoS_ : set set const ap : set set set const SNoCut : set set set const Repl : set (set set) set const abs_SNo : set set var x:set var y:set var z:set var w:set 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 w < x hyp w iIn SNoS_ omega hyp Empty < x + - w claim (?u:set.u iIn omega & (w + eps_ u) <= x) -> w < SNoCut (Repl omega (ap y)) (Repl omega (ap z))