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 omega : set const nat_p : set prop axiom omega_nat_p: !x:set.x iIn omega -> nat_p x const SNo : set prop const eps_ : set set axiom SNo_eps_: !x:set.x iIn omega -> SNo (eps_ x) const SNoLe : set set prop term <= = SNoLe infix <= 2020 2020 const Empty : set const mul_SNo : set set set term * = mul_SNo infix * 2291 2290 axiom nonneg_mul_SNo_Le: !x:set.!y:set.!z:set.SNo x -> Empty <= x -> SNo y -> SNo z -> y <= z -> x * y <= x * z const exp_SNo_nat : set set set const ordsucc : set set axiom mul_SNo_eps_power_2': !x:set.nat_p x -> exp_SNo_nat (ordsucc (ordsucc Empty)) x * eps_ x = ordsucc Empty const real : set 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) -> SNo (exp_SNo_nat (ordsucc (ordsucc Empty)) z) -> Empty <= exp_SNo_nat (ordsucc (ordsucc Empty)) z -> ordsucc Empty <= exp_SNo_nat (ordsucc (ordsucc Empty)) z * x -> ?w:set.w iIn omega & y <= w * 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) claim Empty <= exp_SNo_nat (ordsucc (ordsucc Empty)) z -> ?w:set.w iIn omega & y <= w * x