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 add_SNo : set set set term + = add_SNo infix + 2281 2280 axiom SNo_add_SNo: !x:set.!y:set.SNo x -> SNo y -> SNo (x + y) 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) -> 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