const SNo : set prop const SNoLev : set set const minus_SNo : set set term - = minus_SNo axiom minus_SNo_Lev: !x:set.SNo x -> SNoLev - x = SNoLev x const In : set set prop term iIn = In infix iIn 2000 2000 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 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 = z + x -> SNo z -> SNoLev z iIn SNoLev - x -> z < - x -> SNo - z -> - z + z = Empty -> y < Empty 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 = z + x hyp SNo z hyp SNoLev z iIn SNoLev - x hyp z < - x claim SNo - z -> y < Empty