const In : set set prop term iIn = In infix iIn 2000 2000 const Repl : set (set set) set axiom ReplEq_ext: !x:set.!f:set set.!f2:set set.(!y:set.y iIn x -> f y = f2 y) -> Repl x f = Repl x f2 const SNo : set prop const add_SNo : set set set term + = add_SNo infix + 2281 2280 const SNoCut : set set set const binunion : set set set const SNoL : set set const SNoR : set set axiom add_SNo_eq: !x:set.SNo x -> !y:set.SNo y -> x + y = SNoCut (binunion (Repl (SNoL x) \z:set.z + y) (Repl (SNoL y) (add_SNo x))) (binunion (Repl (SNoR x) \z:set.z + y) (Repl (SNoR y) (add_SNo x))) lemma !x:set.!y:set.(!z:set.z iIn SNoL x -> z + y = y + z) -> (!z:set.z iIn SNoR x -> z + y = y + z) -> (!z:set.z iIn SNoL y -> x + z = z + x) -> (!z:set.z iIn SNoR y -> x + z = z + x) -> Repl (SNoL x) (\z:set.z + y) = Repl (SNoL x) (add_SNo y) -> SNoCut (binunion (Repl (SNoL x) \z:set.z + y) (Repl (SNoL y) (add_SNo x))) (binunion (Repl (SNoR x) \z:set.z + y) (Repl (SNoR y) (add_SNo x))) = SNoCut (binunion (Repl (SNoL y) \z:set.z + x) (Repl (SNoL x) (add_SNo y))) (binunion (Repl (SNoR y) \z:set.z + x) (Repl (SNoR x) (add_SNo y))) const SNoS_ : set set const SNoLev : set set var x:set var y:set hyp SNo x hyp SNo y hyp !z:set.z iIn SNoS_ (SNoLev y) -> x + z = z + x hyp !z:set.z iIn SNoL x -> z + y = y + z hyp !z:set.z iIn SNoR x -> z + y = y + z hyp !z:set.z iIn SNoL y -> x + z = z + x claim (!z:set.z iIn SNoR y -> x + z = z + x) -> x + y = y + x