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 SNoCut : set set set axiom SNoCutP_SNoCut_L: !x:set.!y:set.SNoCutP x y -> !z:set.z iIn x -> z < SNoCut x y const SNoL : set set const binunion : set set set const Repl : set (set set) set const add_SNo : set set set term + = add_SNo infix + 2281 2280 const SNoR : set set var x:set var y:set var z:set hyp SNoCutP (binunion (Repl (SNoL x) \w:set.w + y) (Repl (SNoL y) (add_SNo x))) (binunion (Repl (SNoR x) \w:set.w + y) (Repl (SNoR y) (add_SNo x))) hyp z iIn SNoL y claim x + z iIn binunion (Repl (SNoL x) \w:set.w + y) (Repl (SNoL y) (add_SNo x)) -> (x + z) < SNoCut (binunion (Repl (SNoL x) \w:set.w + y) (Repl (SNoL y) (add_SNo x))) (binunion (Repl (SNoR x) \w:set.w + y) (Repl (SNoR y) (add_SNo x)))