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 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) -> SNo (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 claim SNo (x + y) -> x + y + z = (x + y) + z