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 SNoR : set set const add_SNo : set set set term + = add_SNo infix + 2281 2280 const SNoL : set set const SNoCut : set set set const binunion : set set set lemma !x:set.!y:set.(!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) -> Repl (SNoL y) (add_SNo x) = Repl (SNoL y) (\z:set.z + x) -> 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))) var x:set var y:set 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 hyp !z:set.z iIn SNoR y -> x + z = z + x claim 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)))