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