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