const In : set set prop term iIn = In infix iIn 2000 2000 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 mul_SNo : set set set term * = mul_SNo infix * 2291 2290 axiom mul_SNo_assoc: !x:set.!y:set.!z:set.SNo x -> SNo y -> SNo z -> x * y * z = (x * y) * z const exp_SNo_nat : set set set const ordsucc : set set const Empty : set axiom mul_SNo_eps_power_2': !x:set.nat_p x -> exp_SNo_nat (ordsucc (ordsucc Empty)) x * eps_ x = ordsucc Empty axiom mul_SNo_oneL: !x:set.SNo x -> ordsucc Empty * x = x const SNoLt : set set prop term < = SNoLt infix < 2020 2020 axiom exp_SNo_2_bd: !x:set.nat_p x -> x < exp_SNo_nat (ordsucc (ordsucc Empty)) x const SNoLe : set set prop term <= = SNoLe infix <= 2020 2020 axiom SNoLeLt_tra: !x:set.!y:set.!z:set.SNo x -> SNo y -> SNo z -> x <= y -> y < z -> x < z axiom SNoLt_irref: !x:set.~ x < x var x:set hyp x iIn omega hyp SNo x hyp ordsucc Empty <= eps_ x * x hyp SNo (exp_SNo_nat (ordsucc (ordsucc Empty)) x) claim ~ exp_SNo_nat (ordsucc (ordsucc Empty)) x <= exp_SNo_nat (ordsucc (ordsucc Empty)) x * eps_ x * x