const SNo : set prop const In : set set prop term iIn = In infix iIn 2000 2000 const SNoLev : set set const SNoS_ : set set axiom SNoS_I2: !x:set.!y:set.SNo x -> SNo y -> SNoLev x iIn SNoLev y -> x iIn SNoS_ (SNoLev y) const add_SNo : set set set term + = add_SNo infix + 2281 2280 const minus_SNo : set set term - = minus_SNo const Empty : set const SNoLt : set set prop term < = SNoLt infix < 2020 2020 lemma !x:set.!y:set.!z:set.SNo x -> (!w:set.w iIn SNoS_ (SNoLev x) -> - w + w = Empty) -> SNo - x -> y = - x + z -> SNo z -> SNoLev z iIn SNoLev x -> x < z -> SNo - z -> - z + z = Empty -> Empty < y var x:set var y:set var z:set hyp SNo x hyp !w:set.w iIn SNoS_ (SNoLev x) -> - w + w = Empty hyp SNo - x hyp y = - x + z hyp SNo z hyp SNoLev z iIn SNoLev x hyp x < z claim SNo - z -> Empty < y