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 omega : set axiom nat_p_omega: !x:set.nat_p x -> x iIn omega axiom omega_nat_p: !x:set.x iIn omega -> nat_p x const mul_nat : set set set axiom mul_nat_p: !x:set.nat_p x -> !y:set.nat_p y -> nat_p (mul_nat x y) const mul_SNo : set set set term * = mul_SNo infix * 2291 2290 axiom mul_nat_mul_SNo: !x:set.x iIn omega -> !y:set.y iIn omega -> mul_nat x y = x * y const SNo : set prop axiom omega_SNo: !x:set.x iIn omega -> SNo x const SNoS_ : set set const ordsucc : set set const SNoLt : set set prop term < = SNoLt infix < 2020 2020 axiom SNoS_ordsucc_omega_bdd_above: !x:set.x iIn SNoS_ (ordsucc omega) -> x < omega -> ?y:set.y iIn omega & x < y const real : set const SNoLev : set set const minus_SNo : set set term - = minus_SNo const abs_SNo : set set const add_SNo : set set set term + = add_SNo infix + 2281 2280 const eps_ : set set axiom real_E: !x:set.x iIn real -> !P:prop.(SNo x -> SNoLev x iIn ordsucc omega -> x iIn SNoS_ (ordsucc omega) -> - omega < x -> x < omega -> (!y:set.y iIn SNoS_ omega -> (!z:set.z iIn omega -> abs_SNo (y + - x) < eps_ z) -> y = x) -> (!y:set.y iIn omega -> ?z:set.z iIn SNoS_ omega & (z < x & x < z + eps_ y)) -> P) -> P const exp_SNo_nat : set set set const Empty : set const SNoLe : set set prop term <= = SNoLe infix <= 2020 2020 lemma !x:set.!y:set.!z:set.!w:set.SNo x -> SNo (exp_SNo_nat (ordsucc (ordsucc Empty)) z) -> ordsucc Empty <= exp_SNo_nat (ordsucc (ordsucc Empty)) z * x -> SNo y -> w iIn omega -> y < w -> SNo w -> y <= (w * exp_SNo_nat (ordsucc (ordsucc Empty)) z) * x var x:set var y:set var z:set hyp y iIn real hyp SNo x hyp z iIn omega hyp eps_ z <= x hyp nat_p (exp_SNo_nat (ordsucc (ordsucc Empty)) z) hyp SNo (exp_SNo_nat (ordsucc (ordsucc Empty)) z) hyp Empty <= exp_SNo_nat (ordsucc (ordsucc Empty)) z claim ordsucc Empty <= exp_SNo_nat (ordsucc (ordsucc Empty)) z * x -> ?w:set.w iIn omega & y <= w * x