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 const SNo : set prop const SNoLe : set set prop term <= = SNoLe infix <= 2020 2020 term SNo_max_of = \x:set.\y:set.y iIn x & SNo y & !z:set.z iIn x -> SNo z -> z <= y 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 SNoS_ : set set const SNoLev : set set axiom SNoLev_ind: !p:set prop.(!x:set.SNo x -> (!y:set.y iIn SNoS_ (SNoLev x) -> p y) -> p x) -> !x:set.SNo x -> p x const SNoL : set set const SNoLt : set set prop term < = SNoLt infix < 2020 2020 axiom SNoL_E: !x:set.SNo x -> !y:set.y iIn SNoL x -> !P:prop.(SNo y -> SNoLev y iIn SNoLev x -> y < x -> P) -> P const SNoR : set set lemma !x:set.!y:set.!z:set.SNo x -> SNo y -> (!w:set.w iIn SNoL x -> SNo w -> w <= y) -> SNoLev y iIn SNoLev x -> SNo z -> (!w:set.w iIn SNoS_ (SNoLev z) -> x < w -> (y + w) < x + x -> ?u:set.u iIn SNoR w & y + u = x + x) -> x < z -> (y + z) < x + x -> SNo (x + x) -> ?w:set.w iIn SNoR z & y + w = x + x claim !x:set.!y:set.SNo x -> SNo_max_of (SNoL x) y -> !z:set.SNo z -> x < z -> (y + z) < x + x -> ?w:set.w iIn SNoR z & y + w = x + x