const In : set set prop term iIn = In infix iIn 2000 2000 const SNo : set prop const SNoLt : set set prop term < = SNoLt infix < 2020 2020 term SNoCutP = \x:set.\y:set.(!z:set.z iIn x -> SNo z) & (!z:set.z iIn y -> SNo z) & !z:set.z iIn x -> !w:set.w iIn y -> z < w term Subq = \x:set.\y:set.!z:set.z iIn x -> z iIn y term TransSet = \x:set.!y:set.y iIn x -> Subq y x const add_SNo : set set set term + = add_SNo infix + 2281 2280 const SNoL : set set const SNoR : set set const binunion : set set set const Repl : set (set set) set lemma (!x:set.!y:set.SNo (x + y) & (!z:set.z iIn SNoL x -> (z + y) < x + y) & (!z:set.z iIn SNoR x -> (x + y) < z + y) & (!z:set.z iIn SNoL y -> (x + z) < x + y) & (!z:set.z iIn SNoR y -> (x + y) < x + z) & 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))) -> !P:prop.(SNo (x + y) -> (!z:set.z iIn SNoL x -> (z + y) < x + y) -> (!z:set.z iIn SNoR x -> (x + y) < z + y) -> (!z:set.z iIn SNoL y -> (x + z) < x + y) -> (!z:set.z iIn SNoR y -> (x + y) < x + z) -> P) -> P) -> !x:set.!y:set.SNo x -> SNo y -> SNo (x + y) & (!z:set.z iIn SNoL x -> (z + y) < x + y) & (!z:set.z iIn SNoR x -> (x + y) < z + y) & (!z:set.z iIn SNoL y -> (x + z) < x + y) & (!z:set.z iIn SNoR y -> (x + y) < x + z) & 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))) claim !x:set.!y:set.SNo x -> SNo y -> SNo (x + y) & (!z:set.z iIn SNoL x -> (z + y) < x + y) & (!z:set.z iIn SNoR x -> (x + y) < z + y) & (!z:set.z iIn SNoL y -> (x + z) < x + y) & (!z:set.z iIn SNoR y -> (x + y) < x + z) & 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)))