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 eps_ : set set const SNoS_ : set set axiom SNo_eps_SNoS_omega: !x:set.x iIn omega -> eps_ x iIn SNoS_ omega const real : set axiom SNoS_omega_real: Subq (SNoS_ omega) real const ordsucc : set set const SNoLt : set set prop term < = SNoLt infix < 2020 2020 const Empty : set const mul_SNo : set set set term * = mul_SNo infix * 2291 2290 axiom SNoS_ordsucc_omega_bdd_eps_pos: !x:set.x iIn SNoS_ (ordsucc omega) -> Empty < x -> x < omega -> ?y:set.y iIn omega & eps_ y * x < ordsucc Empty const SNo : set prop 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 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 lemma !x:set.!y:set.x iIn real -> Empty < x -> SNo x -> y iIn omega -> eps_ y * x < ordsucc Empty -> eps_ y iIn real -> ?z:set.z iIn real & x * z = ordsucc Empty claim !x:set.x iIn real -> Empty < x -> ?y:set.y iIn real & x * y = ordsucc Empty