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 ordinal : set prop const omega : set axiom omega_ordinal: ordinal omega const SNo : set prop const Empty : set axiom SNo_0: SNo Empty const nat_p : set prop axiom nat_0: nat_p Empty axiom nat_p_omega: !x:set.nat_p x -> x iIn omega const SNoS_ : set set axiom omega_SNoS_omega: Subq omega (SNoS_ omega) const minus_SNo : set set term - = minus_SNo axiom SNo_minus_SNo: !x:set.SNo x -> SNo - x const SNoLt : set set prop term < = SNoLt infix < 2020 2020 const abs_SNo : set set axiom pos_abs_SNo: !x:set.Empty < x -> abs_SNo x = x axiom abs_SNo_minus: !x:set.SNo x -> abs_SNo - x = abs_SNo x const add_SNo : set set set term + = add_SNo infix + 2281 2280 axiom add_SNo_0L: !x:set.SNo x -> Empty + x = x axiom FalseE: ~ False const eps_ : set set axiom SNo_eps_: !x:set.x iIn omega -> SNo (eps_ x) const SNoLe : set set prop term <= = SNoLe infix <= 2020 2020 axiom SNoLtLe_or: !x:set.!y:set.SNo x -> SNo y -> x < y | y <= x axiom dneg: !P:prop.~ ~ P -> P axiom xm: !P:prop.P | ~ P 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 lemma !x:set.SNo x -> Empty < x -> (!y:set.y iIn SNoS_ omega -> (!z:set.z iIn omega -> abs_SNo (y + - x) < eps_ z) -> y = x) -> (!y:set.y iIn omega -> x < eps_ y) -> Empty != x lemma !x:set.!y:set.!P:prop.!z:set.!w:set.SNo x -> y iIn omega -> (!u:set.u iIn SNoS_ omega -> Empty < u -> u < x -> x < u + eps_ y -> P) -> x < z + eps_ y -> SNo z -> z <= Empty -> ~ P -> w iIn omega -> eps_ w <= x -> SNo (eps_ w) -> x < eps_ w claim !x:set.SNo x -> Empty < x -> (!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)) -> !y:set.y iIn omega -> !P:prop.(!z:set.z iIn SNoS_ omega -> Empty < z -> z < x -> x < z + eps_ y -> P) -> P