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 SNoR : set set const SNoLt : set set prop term < = SNoLt infix < 2020 2020 axiom SNoR_E: !x:set.SNo x -> !y:set.y iIn SNoR x -> !P:prop.(SNo y -> SNoLev y iIn SNoLev x -> x < y -> P) -> P const add_SNo : set set set term + = add_SNo infix + 2281 2280 const SNoL : set set lemma !x:set.!y:set.SNo x -> SNo y -> (!z:set.z iIn SNoS_ (SNoLev x) -> z + y = y + z) -> (!z:set.z iIn SNoS_ (SNoLev y) -> x + z = z + x) -> (!z:set.z iIn SNoL x -> z + y = y + z) -> (!z:set.z iIn SNoR x -> z + y = y + z) -> x + y = y + x lemma !x:set.!y:set.!z:set.SNo x -> (!w:set.w iIn SNoS_ (SNoLev x) -> w + y = y + w) -> SNo z -> SNoLev z iIn SNoLev x -> z iIn SNoS_ (SNoLev x) -> z + y = y + z var x:set var y:set hyp SNo x hyp SNo y hyp !z:set.z iIn SNoS_ (SNoLev x) -> z + y = y + z hyp !z:set.z iIn SNoS_ (SNoLev y) -> x + z = z + x claim (!z:set.z iIn SNoL x -> z + y = y + z) -> x + y = y + x