const In : set set prop term iIn = In infix iIn 2000 2000 const SNo : set prop const SNoLe : set set prop term <= = SNoLe infix <= 2020 2020 term SNo_min_of = \x:set.\y:set.y iIn x & SNo y & !z:set.z iIn x -> SNo z -> y <= z const minus_SNo : set set term - = minus_SNo axiom SNo_minus_SNo: !x:set.SNo x -> SNo - x const SNoR : set set const SNoLev : set set const SNoLt : set set prop term < = SNoLt infix < 2020 2020 axiom SNoR_E: !x:set.SNo x -> !y:set.y iIn SNoR x -> !P:prop.(SNo y -> SNoLev y iIn SNoLev x -> x < y -> P) -> P const add_SNo : set set set term + = add_SNo infix + 2281 2280 const SNoL : set set lemma !x:set.!y:set.!z:set.SNo x -> SNo_min_of (SNoR x) y -> SNo y -> SNo z -> z < x -> (x + x) < y + z -> SNo - x -> ?w:set.w iIn SNoL z & y + w = x + x claim !x:set.!y:set.SNo x -> SNo_min_of (SNoR x) y -> !z:set.SNo z -> z < x -> (x + x) < y + z -> ?w:set.w iIn SNoL z & y + w = x + x