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 SNoLt : set set prop term < = SNoLt infix < 2020 2020 const minus_SNo : set set term - = minus_SNo axiom add_SNo_minus_Lt1: !x:set.!y:set.!z:set.SNo x -> SNo y -> SNo z -> (x + - y) < z -> x < z + y const SNoLe : set set prop term <= = SNoLe infix <= 2020 2020 axiom SNoLtLe_tra: !x:set.!y:set.!z:set.SNo x -> SNo y -> SNo z -> x < y -> y <= z -> 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 SNoS_ : set set const SNoCut : set set set const Repl : set (set set) set const ap : 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 -> 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 x < w hyp w iIn SNoS_ omega hyp Empty < w + - x claim (?u:set.u iIn omega & (x + eps_ u) <= w) -> SNoCut (Repl omega (ap y)) (Repl omega (ap z)) < w