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 SNoLev : set set const SNoS_ : set set 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 var x:set var y:set var z:set hyp SNo y hyp !w:set.w iIn SNoS_ (SNoLev y) -> SNo (x + w) & (!u:set.u iIn SNoL x -> (u + w) < x + w) & (!u:set.u iIn SNoR x -> (x + w) < u + w) & (!u:set.u iIn SNoL w -> (x + u) < x + w) & (!u:set.u iIn SNoR w -> (x + w) < x + u) & SNoCutP (binunion (Repl (SNoL x) \u:set.u + w) (Repl (SNoL w) (add_SNo x))) (binunion (Repl (SNoR x) \u:set.u + w) (Repl (SNoR w) (add_SNo x))) hyp SNo z hyp SNoLev z iIn SNoLev y claim z iIn SNoS_ (SNoLev y) -> SNo (x + z) & (!w:set.w iIn SNoL x -> (w + z) < x + z) & (!w:set.w iIn SNoR x -> (x + z) < w + z) & (!w:set.w iIn SNoL z -> (x + w) < x + z) & (!w:set.w iIn SNoR z -> (x + z) < x + w) & SNoCutP (binunion (Repl (SNoL x) \w:set.w + z) (Repl (SNoL z) (add_SNo x))) (binunion (Repl (SNoR x) \w:set.w + z) (Repl (SNoR z) (add_SNo x)))