const In : set set prop term iIn = In infix iIn 2000 2000 term nIn = \x:set.\y:set.~ x iIn y term Subq = \x:set.\y:set.!z:set.z iIn x -> z iIn y const SNoS_ : set set const omega : set const minus_SNo : set set term - = minus_SNo axiom minus_SNo_SNoS_omega: !x:set.x iIn SNoS_ omega -> - x iIn SNoS_ omega const add_SNo : set set set term + = add_SNo infix + 2281 2280 axiom add_SNo_SNoS_omega: !x:set.x iIn SNoS_ omega -> !y:set.y iIn SNoS_ omega -> x + y iIn SNoS_ omega const SNoLt : set set prop term < = SNoLt infix < 2020 2020 const abs_SNo : set set const eps_ : set set const Empty : set lemma !x:set.!y:set.x iIn SNoS_ omega -> y iIn SNoS_ omega -> (!z:set.z iIn omega -> abs_SNo (y + - x) < eps_ z) -> Empty < y + - x -> ~ y + - x iIn SNoS_ omega const SNo : set prop var x:set var y:set hyp x iIn SNoS_ omega hyp SNo x hyp y iIn SNoS_ omega hyp !z:set.z iIn omega -> abs_SNo (y + - x) < eps_ z hyp SNo y hyp x < y claim ~ Empty < y + - x