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 SNoLt : set set prop term < = SNoLt infix < 2020 2020 const minus_SNo : set set term - = minus_SNo axiom minus_SNo_Lt_contra: !x:set.!y:set.SNo x -> SNo y -> x < y -> - y < - x const add_SNo : set set set term + = add_SNo infix + 2281 2280 axiom minus_add_SNo_distr: !x:set.!y:set.SNo x -> SNo y -> - (x + y) = - x + - y const SNo_max_of : set set prop const SNoL : set set lemma !x:set.!y:set.!z:set.SNo x -> SNo y -> SNo z -> (x + x) < y + z -> SNo - x -> SNo - y -> SNo - z -> SNo (x + x) -> SNo (y + z) -> SNo_max_of (SNoL - x) - y -> - x < - z -> (- y + - z) < - x + - x -> ?w:set.w iIn SNoL z & y + w = x + x var x:set var y:set var z:set hyp SNo x hyp SNo y hyp SNo z hyp z < x hyp (x + x) < y + z hyp SNo - x hyp SNo - y hyp SNo - z hyp SNo (x + x) hyp SNo (y + z) hyp SNo_max_of (SNoL - x) - y claim - x < - z -> ?w:set.w iIn SNoL z & y + w = x + x