const In : set set prop term iIn = In infix iIn 2000 2000 term Subq = \x:set.\y:set.!z:set.z iIn x -> z iIn y const nat_p : set prop const ordsucc : set set const Empty : set axiom nat_2: nat_p (ordsucc (ordsucc Empty)) const omega : set axiom omega_nat_p: !x:set.x iIn omega -> nat_p x const exp_SNo_nat : set set set axiom nat_exp_SNo_nat: !x:set.nat_p x -> !y:set.nat_p y -> nat_p (exp_SNo_nat x y) const real : set const SNo : set prop const SNoLe : set set prop term <= = SNoLe infix <= 2020 2020 const eps_ : set set const mul_SNo : set set set term * = mul_SNo infix * 2291 2290 lemma !x:set.!y:set.!z:set.y iIn real -> SNo x -> z iIn omega -> eps_ z <= x -> nat_p (exp_SNo_nat (ordsucc (ordsucc Empty)) z) -> ?w:set.w iIn omega & y <= w * x const SNoS_ : set set const SNoLt : set set prop term < = SNoLt infix < 2020 2020 const abs_SNo : set set const add_SNo : set set set term + = add_SNo infix + 2281 2280 const minus_SNo : set set term - = minus_SNo var x:set var y:set hyp y iIn real hyp Empty < x hyp SNo x hyp !z:set.z iIn SNoS_ omega -> (!w:set.w iIn omega -> abs_SNo (z + - x) < eps_ w) -> z = x claim (?z:set.z iIn omega & eps_ z <= x) -> ?z:set.z iIn omega & y <= z * x