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 SNoR : set set const SNoS_ : set set const SNoLev : set set axiom SNoR_SNoS_: !x:set.Subq (SNoR x) (SNoS_ (SNoLev x)) const SNoLt : set set prop term < = SNoLt infix < 2020 2020 axiom SNoLt_tra: !x:set.!y:set.!z:set.SNo x -> SNo y -> SNo z -> x < y -> y < z -> x < z 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 y -> 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) -> SNo (y + z) -> (!w:set.w iIn SNoR y -> ~ (w + z) <= x + x) -> (!w:set.w iIn SNoL x -> ~ (y + z) <= w + x) -> (!w:set.w iIn SNoR z -> (y + w) < x + x -> ?u:set.u iIn SNoR z & y + u = x + x) -> ?w:set.w iIn SNoR z & y + w = x + x lemma !x:set.!y:set.!z:set.!w:set.SNo x -> SNo z -> (!u:set.u iIn SNoS_ (SNoLev z) -> x < u -> (y + u) < x + x -> ?v:set.v iIn SNoR u & y + v = x + x) -> x < z -> w iIn SNoR z -> (y + w) < x + x -> SNo w -> SNoLev w iIn SNoLev z -> z < w -> (?u:set.u iIn SNoR w & y + u = x + x) -> ?u:set.u iIn SNoR z & y + u = x + x var x:set var y:set var z:set hyp SNo x hyp SNo y hyp !w:set.w iIn SNoL x -> SNo w -> w <= y hyp SNo z hyp !w:set.w iIn SNoS_ (SNoLev z) -> x < w -> (y + w) < x + x -> ?u:set.u iIn SNoR w & y + u = x + x hyp x < z hyp (y + z) < x + x hyp SNo (x + x) hyp SNo (y + z) hyp !w:set.w iIn SNoR y -> ~ (w + z) <= x + x claim (!w:set.w iIn SNoL x -> ~ (y + z) <= w + x) -> ?w:set.w iIn SNoR z & y + w = x + x