const binunion : set set set axiom binunion_com: !x:set.!y:set.binunion x y = binunion y x const Repl : set (set set) set const SNoR : set set const add_SNo : set set set term + = add_SNo infix + 2281 2280 const SNoL : set set const In : set set prop term iIn = In infix iIn 2000 2000 const SNoCut : set set set var x:set var y:set hyp !z:set.z iIn SNoR y -> x + z = z + x hyp Repl (SNoL x) (\z:set.z + y) = Repl (SNoL x) (add_SNo y) hyp Repl (SNoL y) (add_SNo x) = Repl (SNoL y) \z:set.z + x hyp Repl (SNoR x) (\z:set.z + y) = Repl (SNoR x) (add_SNo y) claim Repl (SNoR y) (add_SNo x) = Repl (SNoR 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)))