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