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 SNo : set prop const ordsucc : set set const Empty : set axiom SNo_2: SNo (ordsucc (ordsucc Empty)) const omega : set const nat_p : set prop axiom omega_nat_p: !x:set.x iIn omega -> nat_p x const exp_SNo_nat : set set set axiom SNo_exp_SNo_nat: !x:set.SNo x -> !y:set.nat_p y -> SNo (exp_SNo_nat x y) const real : set 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) -> SNo (exp_SNo_nat (ordsucc (ordsucc Empty)) z) -> ?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 claim nat_p (exp_SNo_nat (ordsucc (ordsucc Empty)) z) -> ?w:set.w iIn omega & y <= w * x