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 SNoLt : set set prop term < = SNoLt infix < 2020 2020 axiom add_SNo_Lt2: !x:set.!y:set.!z:set.SNo x -> SNo y -> SNo z -> y < z -> (x + y) < x + z axiom add_SNo_Le1: !x:set.!y:set.!z:set.SNo x -> SNo y -> SNo z -> x <= z -> (x + y) <= z + y axiom SNoLe_tra: !x:set.!y:set.!z:set.SNo x -> SNo y -> SNo z -> x <= y -> y <= z -> x <= z axiom SNoLtLe_tra: !x:set.!y:set.!z:set.SNo x -> SNo y -> SNo z -> x < y -> y <= z -> x < z axiom SNoLt_irref: !x:set.~ x < x const SNoL : set set const SNoLev : set set 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 SNoS_ : set set 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) -> 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 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 SNoLev y iIn SNoLev x 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) claim (!w:set.w iIn SNoR y -> ~ (w + z) <= x + x) -> ?w:set.w iIn SNoR z & y + w = x + x