const Pi : set (set set) set term setexp = \x:set.\y:set.Pi y \z:set.x const SNo : set prop 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 In : set set prop term iIn = In infix iIn 2000 2000 const ap : set set set const Sigma : set (set set) set axiom beta: !x:set.!f:set set.!y:set.y iIn x -> ap (Sigma x f) y = f y const omega : set var x:set var y:set var z:set hyp !w:set.w iIn omega -> SNo (ap x w) hyp y iIn omega hyp !w:set.w iIn y -> ap x w < ap x y hyp z iIn y claim z iIn omega -> - ap x y < ap (Sigma omega \w:set.- ap x w) z