const Pi : set (set set) set term setexp = \x:set.\y:set.Pi y \z:set.x const SNoLt : set set prop term < = SNoLt infix < 2020 2020 const add_SNo : set set set term + = add_SNo infix + 2281 2280 const minus_SNo : set set term - = minus_SNo const ap : set set set const eps_ : set set const In : set set prop term iIn = In infix iIn 2000 2000 const Sigma : set (set set) set const omega : set var x:set var y:set var z:set hyp z iIn omega hyp (- ap y z + - eps_ z) < x & x < - ap y z & !w:set.w iIn z -> - ap y z < ap (Sigma omega \u:set.- ap y u) w claim ap (Sigma omega \w:set.- ap y w) z = - ap y z -> (ap (Sigma omega \w:set.- ap y w) z + - eps_ z) < x & x < ap (Sigma omega \w:set.- ap y w) z & !w:set.w iIn z -> ap (Sigma omega \u:set.- ap y u) z < ap (Sigma omega \u:set.- ap y u) w