const SNo : set prop const add_SNo : set set set term + = add_SNo infix + 2281 2280 axiom SNo_add_SNo: !x:set.!y:set.SNo x -> SNo y -> SNo (x + y) const In : set set prop term iIn = In infix iIn 2000 2000 const SNoLev : set set const SNoLt : set set prop term < = SNoLt infix < 2020 2020 const SNoR : set set axiom SNoR_I: !x:set.SNo x -> !y:set.SNo y -> SNoLev y iIn SNoLev x -> x < y -> y iIn SNoR x const SNoL : set set axiom SNoL_I: !x:set.SNo x -> !y:set.SNo y -> SNoLev y iIn SNoLev x -> y < x -> y iIn SNoL x axiom SNoLt_tra: !x:set.!y:set.!z:set.SNo x -> SNo y -> SNo z -> x < y -> y < z -> x < z const binintersect : set set set axiom binintersectE: !x:set.!y:set.!z:set.z iIn binintersect x y -> z iIn x & z iIn y const SNoEq_ : set set set prop const nIn : set set prop axiom SNoLtE: !x:set.!y:set.SNo x -> SNo y -> x < y -> !P:prop.(!z:set.SNo z -> SNoLev z iIn binintersect (SNoLev x) (SNoLev y) -> SNoEq_ (SNoLev z) z x -> SNoEq_ (SNoLev z) z y -> x < z -> z < y -> nIn (SNoLev z) x -> SNoLev z iIn y -> P) -> (SNoLev x iIn SNoLev y -> SNoEq_ (SNoLev x) x y -> SNoLev x iIn y -> P) -> (SNoLev y iIn SNoLev x -> SNoEq_ (SNoLev y) x y -> nIn (SNoLev y) x -> P) -> P 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.!z:set.SNo x -> SNo y -> SNo z -> x < z -> (x + y) < z + y