const In : set set prop term iIn = In infix iIn 2000 2000 term Subq = \x:set.\y:set.!z:set.z iIn x -> z iIn y 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 SNoS_ : set set const SNoLev : set set lemma !x:set.!y:set.!z:set.SNo x -> SNo y -> SNo z -> (!w:set.w iIn SNoS_ (SNoLev x) -> w + y + z = (w + y) + z) -> (!w:set.w iIn SNoS_ (SNoLev y) -> x + w + z = (x + w) + z) -> (!w:set.w iIn SNoS_ (SNoLev z) -> x + y + w = (x + y) + w) -> SNo (x + y) -> SNo (y + z) -> SNoCutP (binunion (Repl (SNoL x) \w:set.w + y + z) (Repl (SNoL (y + z)) (add_SNo x))) (binunion (Repl (SNoR x) \w:set.w + y + z) (Repl (SNoR (y + z)) (add_SNo x))) -> SNoCut (binunion (Repl (SNoL x) \w:set.w + y + z) (Repl (SNoL (y + z)) (add_SNo x))) (binunion (Repl (SNoR x) \w:set.w + y + z) (Repl (SNoR (y + z)) (add_SNo x))) = SNoCut (binunion (Repl (SNoL (x + y)) \w:set.w + z) (Repl (SNoL z) (add_SNo (x + y)))) (binunion (Repl (SNoR (x + y)) \w:set.w + z) (Repl (SNoR z) (add_SNo (x + y)))) var x:set var y:set var z:set hyp SNo x hyp SNo y hyp SNo z hyp !w:set.w iIn SNoS_ (SNoLev x) -> w + y + z = (w + y) + z hyp !w:set.w iIn SNoS_ (SNoLev y) -> x + w + z = (x + w) + z hyp !w:set.w iIn SNoS_ (SNoLev z) -> x + y + w = (x + y) + w hyp SNo (x + y) claim SNo (y + z) -> x + y + z = (x + y) + z