const Pi : set (set set) set term setexp = \x:set.\y:set.Pi y \z:set.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 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 eps_ : set set lemma !x:set.!y:set.!z:set.z iIn omega -> (- 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) -> 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 const SNo : set prop var x:set var y:set var z:set hyp SNo x hyp !w:set.w iIn omega -> ap y w < - x & - x < ap y w + eps_ w & !u:set.u iIn w -> ap y u < ap y w hyp !w:set.w iIn omega -> SNo (ap y w) hyp z iIn omega claim (- 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) -> (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