const In : set set prop term iIn = In infix iIn 2000 2000 term nIn = \x:set.\y:set.~ x iIn y 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 Empty : set const add_SNo : set set set term + = add_SNo infix + 2281 2280 const minus_SNo : set set term - = minus_SNo const nat_primrec : set (set set set) set set const Eps_i : (set prop) set const SNoS_ : set set const omega : set const eps_ : set set const ordsucc : set set const SNoLe : set set prop term <= = SNoLe infix <= 2020 2020 const SNo : set prop const abs_SNo : set set var x:set var y:set hyp SNo x hyp !z:set.z iIn SNoS_ omega -> (!w:set.w iIn omega -> abs_SNo (z + - x) < eps_ w) -> z = x hyp nat_primrec (Eps_i \z:set.z iIn SNoS_ omega & z < x & x < z + eps_ Empty) (\z:set.\w:set.Eps_i \u:set.u iIn SNoS_ omega & u < x & x < u + eps_ (ordsucc z) & w < u) y iIn SNoS_ omega hyp nat_primrec (Eps_i \z:set.z iIn SNoS_ omega & z < x & x < z + eps_ Empty) (\z:set.\w:set.Eps_i \u:set.u iIn SNoS_ omega & u < x & x < u + eps_ (ordsucc z) & w < u) y < x hyp SNo (nat_primrec (Eps_i \z:set.z iIn SNoS_ omega & z < x & x < z + eps_ Empty) (\z:set.\w:set.Eps_i \u:set.u iIn SNoS_ omega & u < x & x < u + eps_ (ordsucc z) & w < u) y) hyp SNo (x + - nat_primrec (Eps_i \z:set.z iIn SNoS_ omega & z < x & x < z + eps_ Empty) (\z:set.\w:set.Eps_i \u:set.u iIn SNoS_ omega & u < x & x < u + eps_ (ordsucc z) & w < u) y) hyp ~ ?z:set.z iIn omega & eps_ z <= x + - nat_primrec (Eps_i \w:set.w iIn SNoS_ omega & w < x & x < w + eps_ Empty) (\w:set.\u:set.Eps_i \v:set.v iIn SNoS_ omega & v < x & x < v + eps_ (ordsucc w) & u < v) y hyp Empty < x + - nat_primrec (Eps_i \z:set.z iIn SNoS_ omega & z < x & x < z + eps_ Empty) (\z:set.\w:set.Eps_i \u:set.u iIn SNoS_ omega & u < x & x < u + eps_ (ordsucc z) & w < u) y claim nat_primrec (Eps_i \z:set.z iIn SNoS_ omega & z < x & x < z + eps_ Empty) (\z:set.\w:set.Eps_i \u:set.u iIn SNoS_ omega & u < x & x < u + eps_ (ordsucc z) & w < u) y = x -> x < x