const In : set set prop term iIn = In infix iIn 2000 2000 const omega : set const eps_ : set set const SNoS_ : set set axiom SNo_eps_SNoS_omega: !x:set.x iIn omega -> eps_ 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 minus_SNo : set set term - = minus_SNo axiom minus_SNo_SNoS_omega: !x:set.x iIn SNoS_ omega -> - x iIn SNoS_ omega const SNo : set prop const SNoLt : set set prop term < = SNoLt infix < 2020 2020 axiom minus_SNo_Lt_contra: !x:set.!y:set.SNo x -> SNo y -> x < y -> - y < - x axiom SNo_eps_: !x:set.x iIn omega -> SNo (eps_ x) axiom SNo_minus_SNo: !x:set.SNo x -> SNo - x axiom add_SNo_minus_R2': !x:set.!y:set.SNo x -> SNo y -> (x + - y) + y = x axiom minus_add_SNo_distr: !x:set.!y:set.SNo x -> SNo y -> - (x + y) = - x + - y var x:set var y:set var z:set hyp SNo x hyp y iIn omega hyp z iIn SNoS_ omega hyp z < x hyp x < z + eps_ y hyp SNo z claim SNo (z + eps_ y) -> ?w:set.w iIn SNoS_ omega & (w < - x & - x < w + eps_ y)