const In : set set prop term iIn = In infix iIn 2000 2000 const omega : set const SNo : set prop const eps_ : set set axiom SNo_eps_: !x:set.x iIn omega -> SNo (eps_ x) const nat_p : set prop axiom omega_nat_p: !x:set.x iIn omega -> nat_p x const SNoLt : set set prop term < = SNoLt infix < 2020 2020 const Empty : set axiom SNo_eps_pos: !x:set.x iIn omega -> Empty < eps_ x const SNoLe : set set prop term <= = SNoLe infix <= 2020 2020 axiom SNoLtLe: !x:set.!y:set.x < y -> x <= y const ordsucc : set set axiom SNo_1: SNo (ordsucc Empty) axiom SNo_2: SNo (ordsucc (ordsucc Empty)) 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) axiom SNoLt_1_2: ordsucc Empty < ordsucc (ordsucc Empty) axiom exp_SNo_1_bd: !x:set.SNo x -> ordsucc Empty <= x -> !y:set.nat_p y -> ordsucc Empty <= exp_SNo_nat x y 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 axiom mul_SNo_eps_power_2: !x:set.nat_p x -> eps_ x * exp_SNo_nat (ordsucc (ordsucc Empty)) x = ordsucc Empty axiom mul_SNo_oneR: !x:set.SNo x -> x * ordsucc Empty = x claim !x:set.x iIn omega -> eps_ x <= ordsucc Empty