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 SNoS_ : set set const minus_SNo : set set term - = minus_SNo axiom minus_SNo_SNoS_: !x:set.ordinal x -> !y:set.y iIn SNoS_ x -> - y iIn SNoS_ x const SNo : set prop axiom minus_SNo_invol: !x:set.SNo x -> - - x = x const SNoLt : set set prop term < = SNoLt infix < 2020 2020 axiom SNoLt_irref: !x:set.~ x < x axiom SNo_omega: SNo omega const abs_SNo : set set const add_SNo : set set set term + = add_SNo infix + 2281 2280 const eps_ : set set axiom minus_SNo_prereal_1: !x:set.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 SNoS_ omega -> (!z:set.z iIn omega -> abs_SNo (y + - - x) < eps_ z) -> y = - x const real : set axiom real_I: !x:set.x iIn SNoS_ (ordsucc omega) -> x != omega -> x != - omega -> (!y:set.y iIn SNoS_ omega -> (!z:set.z iIn omega -> abs_SNo (y + - x) < eps_ z) -> y = x) -> x iIn real const SNoLev : 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 claim !x:set.x iIn real -> - x iIn real