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 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) -> SNo (y * z) -> SNo ((x * y) * z) -> x * y * z = (x * y) * z var x:set var y:set var z:set hyp SNo x hyp SNo y hyp SNo z hyp !w:set.w iIn SNoS_ (SNoLev x) -> w * y * z = (w * y) * z hyp !w:set.w iIn SNoS_ (SNoLev y) -> x * w * z = (x * w) * z hyp !w:set.w iIn SNoS_ (SNoLev z) -> x * y * w = (x * y) * w hyp !w:set.w iIn SNoS_ (SNoLev x) -> !u:set.u iIn SNoS_ (SNoLev y) -> w * u * z = (w * u) * z hyp !w:set.w iIn SNoS_ (SNoLev x) -> !u:set.u iIn SNoS_ (SNoLev z) -> w * y * u = (w * y) * u hyp !w:set.w iIn SNoS_ (SNoLev y) -> !u:set.u iIn SNoS_ (SNoLev z) -> x * w * u = (x * w) * u hyp !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 hyp SNo (x * y) hyp SNo (y * z) claim SNo (x * y * z) -> x * y * z = (x * y) * z