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