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