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 SNoLt : set set prop term < = SNoLt infix < 2020 2020 term SNoCutP = \x:set.\y:set.(!z:set.z iIn x -> SNo z) & (!z:set.z iIn y -> SNo z) & !z:set.z iIn x -> !w:set.w iIn y -> z < w const omega : set const ordsucc : set set axiom omega_ordsucc: !x:set.x iIn omega -> ordsucc x iIn omega const eps_ : set set axiom SNo_eps_: !x:set.x iIn omega -> SNo (eps_ x) const Empty : set const Sep : set (set prop) set const SNoS_ : set set const mul_SNo : set set set term * = mul_SNo infix * 2291 2290 const SNoCut : set set set const add_SNo : set set set term + = add_SNo infix + 2281 2280 const abs_SNo : set set const minus_SNo : set set term - = minus_SNo lemma !x:set.!y:set.!z:set.Empty < x -> x < ordsucc Empty -> SNo x -> (!w:set.w iIn Sep (SNoS_ omega) (\u:set.x * u < ordsucc Empty) -> w < SNoCut (Sep (SNoS_ omega) \u:set.x * u < ordsucc Empty) (Sep (SNoS_ omega) \u:set.ordsucc Empty < x * u)) -> (!w:set.w iIn Sep (SNoS_ omega) (\u:set.ordsucc Empty < x * u) -> SNoCut (Sep (SNoS_ omega) \u:set.x * u < ordsucc Empty) (Sep (SNoS_ omega) \u:set.ordsucc Empty < x * u) < w) -> SNo (SNoCut (Sep (SNoS_ omega) \w:set.x * w < ordsucc Empty) (Sep (SNoS_ omega) \w:set.ordsucc Empty < x * w)) -> SNo (x * SNoCut (Sep (SNoS_ omega) \w:set.x * w < ordsucc Empty) (Sep (SNoS_ omega) \w:set.ordsucc Empty < x * w)) -> SNo (x * SNoCut (Sep (SNoS_ omega) \w:set.x * w < ordsucc Empty) (Sep (SNoS_ omega) \w:set.ordsucc Empty < x * w)) -> y iIn omega -> z iIn SNoS_ omega -> z < SNoCut (Sep (SNoS_ omega) \w:set.x * w < ordsucc Empty) (Sep (SNoS_ omega) \w:set.ordsucc Empty < x * w) -> SNoCut (Sep (SNoS_ omega) \w:set.x * w < ordsucc Empty) (Sep (SNoS_ omega) \w:set.ordsucc Empty < x * w) < z + eps_ (ordsucc y) -> SNo z -> SNo (x * z) -> SNo (eps_ (ordsucc y)) -> abs_SNo (ordsucc Empty + - x * SNoCut (Sep (SNoS_ omega) \w:set.x * w < ordsucc Empty) (Sep (SNoS_ omega) \w:set.ordsucc Empty < x * w)) < eps_ y var x:set var y:set var z:set hyp Empty < x hyp x < ordsucc Empty hyp SNo x hyp !w:set.w iIn Sep (SNoS_ omega) (\u:set.x * u < ordsucc Empty) -> w < SNoCut (Sep (SNoS_ omega) \u:set.x * u < ordsucc Empty) (Sep (SNoS_ omega) \u:set.ordsucc Empty < x * u) hyp !w:set.w iIn Sep (SNoS_ omega) (\u:set.ordsucc Empty < x * u) -> SNoCut (Sep (SNoS_ omega) \u:set.x * u < ordsucc Empty) (Sep (SNoS_ omega) \u:set.ordsucc Empty < x * u) < w hyp SNo (SNoCut (Sep (SNoS_ omega) \w:set.x * w < ordsucc Empty) (Sep (SNoS_ omega) \w:set.ordsucc Empty < x * w)) hyp SNo (x * SNoCut (Sep (SNoS_ omega) \w:set.x * w < ordsucc Empty) (Sep (SNoS_ omega) \w:set.ordsucc Empty < x * w)) hyp SNo (x * SNoCut (Sep (SNoS_ omega) \w:set.x * w < ordsucc Empty) (Sep (SNoS_ omega) \w:set.ordsucc Empty < x * w)) hyp y iIn omega hyp z iIn SNoS_ omega hyp z < SNoCut (Sep (SNoS_ omega) \w:set.x * w < ordsucc Empty) (Sep (SNoS_ omega) \w:set.ordsucc Empty < x * w) hyp SNoCut (Sep (SNoS_ omega) \w:set.x * w < ordsucc Empty) (Sep (SNoS_ omega) \w:set.ordsucc Empty < x * w) < z + eps_ (ordsucc y) hyp SNo z claim SNo (x * z) -> abs_SNo (ordsucc Empty + - x * SNoCut (Sep (SNoS_ omega) \w:set.x * w < ordsucc Empty) (Sep (SNoS_ omega) \w:set.ordsucc Empty < x * w)) < eps_ y