const ordinal : set prop const ordsucc : set set const omega : set axiom ordsucc_omega_ordinal: ordinal (ordsucc omega) const In : set set prop term iIn = In infix iIn 2000 2000 const SNo : set prop axiom omega_SNo: !x:set.x iIn omega -> SNo x const SNoS_ : set set const SNoLt : set set prop term < = SNoLt infix < 2020 2020 axiom SNoS_ordsucc_omega_bdd_above: !x:set.x iIn SNoS_ (ordsucc omega) -> x < omega -> ?y:set.y iIn omega & x < y const SNoLev : set set const SNo_ : set set prop axiom SNoS_E2: !x:set.ordinal x -> !y:set.y iIn SNoS_ x -> !P:prop.(SNoLev y iIn x -> ordinal (SNoLev y) -> SNo y -> SNo_ (SNoLev y) y -> P) -> P const mul_SNo : set set set term * = mul_SNo infix * 2291 2290 const eps_ : set set const Empty : set lemma !x:set.!y:set.SNo x -> y iIn omega -> x < y -> SNo y -> ?z:set.z iIn omega & eps_ z * x < ordsucc Empty claim !x:set.x iIn SNoS_ (ordsucc omega) -> Empty < x -> x < omega -> ?y:set.y iIn omega & eps_ y * x < ordsucc Empty