const SNo : set prop const mul_SNo : set set set term * = mul_SNo infix * 2291 2290 axiom SNo_mul_SNo: !x:set.!y:set.SNo x -> SNo y -> SNo (x * y) const In : set set prop term iIn = In infix iIn 2000 2000 const SNoS_ : set set const SNoLev : set set axiom SNoLev_ind3: !P:set set set prop.(!x:set.!y:set.!z:set.SNo x -> SNo y -> SNo z -> (!w:set.w iIn SNoS_ (SNoLev x) -> P w y z) -> (!w:set.w iIn SNoS_ (SNoLev y) -> P x w z) -> (!w:set.w iIn SNoS_ (SNoLev z) -> P x y w) -> (!w:set.w iIn SNoS_ (SNoLev x) -> !u:set.u iIn SNoS_ (SNoLev y) -> P w u z) -> (!w:set.w iIn SNoS_ (SNoLev x) -> !u:set.u iIn SNoS_ (SNoLev z) -> P w y u) -> (!w:set.w iIn SNoS_ (SNoLev y) -> !u:set.u iIn SNoS_ (SNoLev z) -> P x w u) -> (!w:set.w iIn SNoS_ (SNoLev x) -> !u:set.u iIn SNoS_ (SNoLev y) -> !v:set.v iIn SNoS_ (SNoLev z) -> P w u v) -> P x y z) -> !x:set.!y:set.!z:set.SNo x -> SNo y -> SNo z -> P x y z lemma !x:set.!y:set.!z:set.SNo x -> SNo y -> SNo z -> (!w:set.w iIn SNoS_ (SNoLev x) -> w * y * z = (w * y) * z) -> (!w:set.w iIn SNoS_ (SNoLev y) -> x * w * z = (x * w) * z) -> (!w:set.w iIn SNoS_ (SNoLev z) -> x * y * w = (x * y) * w) -> (!w:set.w iIn SNoS_ (SNoLev x) -> !u:set.u iIn SNoS_ (SNoLev y) -> w * u * z = (w * u) * z) -> (!w:set.w iIn SNoS_ (SNoLev x) -> !u:set.u iIn SNoS_ (SNoLev z) -> w * y * u = (w * y) * u) -> (!w:set.w iIn SNoS_ (SNoLev y) -> !u:set.u iIn SNoS_ (SNoLev z) -> x * w * u = (x * w) * u) -> (!w:set.w iIn SNoS_ (SNoLev x) -> !u:set.u iIn SNoS_ (SNoLev y) -> !v:set.v iIn SNoS_ (SNoLev z) -> w * u * v = (w * u) * v) -> SNo (x * y) -> x * y * z = (x * y) * z claim !x:set.!y:set.!z:set.SNo x -> SNo y -> SNo z -> x * y * z = (x * y) * z