const In : set set prop term iIn = In infix iIn 2000 2000 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 term ordinal = \x:set.TransSet x & !y:set.y iIn x -> TransSet y 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 SNoL : set set const SNoLe : set set prop term <= = SNoLe infix <= 2020 2020 lemma !x:set.!y:set.SNo x -> SNo y -> SNo (x + y) -> !z:set.z iIn SNoL (x + y) -> (?w:set.w iIn SNoL x & z <= w + y) | ?w:set.w iIn SNoL y & z <= x + w claim !x:set.!y:set.SNo x -> SNo y -> !z:set.z iIn SNoL (x + y) -> (?w:set.w iIn SNoL x & z <= w + y) | ?w:set.w iIn SNoL y & z <= x + w