const SNo_rec2 : (set set (set set set) set) set set set const SNoCut : set set set const binunion : set set set const Repl : set (set set) set const SNoL : set set const SNoR : set set term add_SNo = SNo_rec2 \x:set.\y:set.\g:set set set.SNoCut (binunion (Repl (SNoL x) \z:set.g z y) (Repl (SNoL y) (g x))) (binunion (Repl (SNoR x) \z:set.g z y) (Repl (SNoR y) (g x))) term + = add_SNo infix + 2281 2280 const SNo : set prop const In : set set prop term iIn = In infix iIn 2000 2000 const SNoS_ : set set const SNoLev : set set axiom SNo_rec2_eq: !P:set set (set set set) set.(!x:set.SNo x -> !y:set.SNo y -> !g:set set set.!h:set set set.(!z:set.z iIn SNoS_ (SNoLev x) -> !w:set.SNo w -> g z w = h z w) -> (!z:set.z iIn SNoS_ (SNoLev y) -> g x z = h x z) -> P x y g = P x y h) -> !x:set.SNo x -> !y:set.SNo y -> SNo_rec2 P x y = P x y (SNo_rec2 P) claim (!x:set.SNo x -> !y:set.SNo y -> !g:set set set.!h:set set set.(!z:set.z iIn SNoS_ (SNoLev x) -> !w:set.SNo w -> g z w = h z w) -> (!z:set.z iIn SNoS_ (SNoLev y) -> g x z = h x z) -> SNoCut (binunion (Repl (SNoL x) \z:set.g z y) (Repl (SNoL y) (g x))) (binunion (Repl (SNoR x) \z:set.g z y) (Repl (SNoR y) (g x))) = SNoCut (binunion (Repl (SNoL x) \z:set.h z y) (Repl (SNoL y) (h x))) (binunion (Repl (SNoR x) \z:set.h z y) (Repl (SNoR y) (h x)))) -> !x:set.SNo x -> !y:set.SNo y -> SNo_rec2 (\z:set.\w:set.\g:set set set.SNoCut (binunion (Repl (SNoL z) \u:set.g u w) (Repl (SNoL w) (g z))) (binunion (Repl (SNoR z) \u:set.g u w) (Repl (SNoR w) (g z)))) x y = SNoCut (binunion (Repl (SNoL x) \z:set.SNo_rec2 (\w:set.\u:set.\g:set set set.SNoCut (binunion (Repl (SNoL w) \v:set.g v u) (Repl (SNoL u) (g w))) (binunion (Repl (SNoR w) \v:set.g v u) (Repl (SNoR u) (g w)))) z y) (Repl (SNoL y) (SNo_rec2 (\z:set.\w:set.\g:set set set.SNoCut (binunion (Repl (SNoL z) \u:set.g u w) (Repl (SNoL w) (g z))) (binunion (Repl (SNoR z) \u:set.g u w) (Repl (SNoR w) (g z)))) x))) (binunion (Repl (SNoR x) \z:set.SNo_rec2 (\w:set.\u:set.\g:set set set.SNoCut (binunion (Repl (SNoL w) \v:set.g v u) (Repl (SNoL u) (g w))) (binunion (Repl (SNoR w) \v:set.g v u) (Repl (SNoR u) (g w)))) z y) (Repl (SNoR y) (SNo_rec2 (\z:set.\w:set.\g:set set set.SNoCut (binunion (Repl (SNoL z) \u:set.g u w) (Repl (SNoL w) (g z))) (binunion (Repl (SNoR z) \u:set.g u w) (Repl (SNoR w) (g z)))) x)))