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 nat_p : set prop const Empty : set axiom nat_0: nat_p Empty const omega : set 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 SNo : set prop const minus_SNo : set set term - = minus_SNo axiom SNo_minus_SNo: !x:set.SNo x -> SNo - x const eps_ : set set axiom SNo_eps_: !x:set.x iIn omega -> SNo (eps_ x) axiom FalseE: ~ False const SNoLt : set set prop term < = SNoLt infix < 2020 2020 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 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 dneg: !P:prop.~ ~ P -> P const real : set const SNoLev : set set const ordsucc : set set 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 const mul_SNo : set set set term * = mul_SNo infix * 2291 2290 lemma !x:set.!y:set.y iIn real -> Empty < x -> SNo x -> (!z:set.z iIn SNoS_ omega -> (!w:set.w iIn omega -> abs_SNo (z + - x) < eps_ w) -> z = x) -> (?z:set.z iIn omega & eps_ z <= x) -> ?z:set.z iIn omega & y <= z * x lemma !x:set.Empty < x -> SNo 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 & eps_ y <= x) -> Empty != x claim !x:set.x iIn real -> !y:set.y iIn real -> Empty < x -> Empty <= y -> ?z:set.z iIn omega & y <= z * x