const SNo : set prop const add_SNo : set set set term + = add_SNo infix + 2281 2280 const In : set set prop term iIn = In infix iIn 2000 2000 const SNoL : set set const SNoLt : set set prop term < = SNoLt infix < 2020 2020 const SNoR : set set const SNoCutP : set set prop const binunion : set set set const Repl : set (set set) set axiom add_SNo_prop1: !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)