const In : set set prop term iIn = In infix iIn 2000 2000 term Subq = \x:set.\y:set.!z:set.z iIn x -> z iIn y const SNo : set prop const minus_SNo : set set term - = minus_SNo axiom SNo_minus_SNo: !x:set.SNo x -> SNo - x const SNoS_ : set set const SNoLev : set set axiom SNoLev_ind2: !r:set set prop.(!x:set.!y:set.SNo x -> SNo y -> (!z:set.z iIn SNoS_ (SNoLev x) -> r z y) -> (!z:set.z iIn SNoS_ (SNoLev y) -> r x z) -> (!z:set.z iIn SNoS_ (SNoLev x) -> !w:set.w iIn SNoS_ (SNoLev y) -> r z w) -> r x y) -> !x:set.!y:set.SNo x -> SNo y -> r x y const mul_SNo : set set set term * = mul_SNo infix * 2291 2290 lemma !x:set.!y:set.SNo x -> SNo y -> (!z:set.z iIn SNoS_ (SNoLev x) -> (- z) * y = - z * y) -> (!z:set.z iIn SNoS_ (SNoLev y) -> (- x) * z = - x * z) -> (!z:set.z iIn SNoS_ (SNoLev x) -> !w:set.w iIn SNoS_ (SNoLev y) -> (- z) * w = - z * w) -> SNo - x -> (- x) * y = - x * y claim !x:set.!y:set.SNo x -> SNo y -> (- x) * y = - x * y