const In : set set prop term iIn = In infix iIn 2000 2000 term nIn = \x:set.\y:set.~ x iIn y 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 eps_ : set set axiom SNo_eps_: !x:set.x iIn omega -> SNo (eps_ x) const SNoLt : set set prop term < = SNoLt infix < 2020 2020 const Empty : set const abs_SNo : set set axiom pos_abs_SNo: !x:set.Empty < x -> abs_SNo x = x const nat_p : set prop axiom omega_nat_p: !x:set.x iIn omega -> nat_p x const ordsucc : set set axiom ordinal_ordsucc: !x:set.ordinal x -> ordinal (ordsucc x) axiom ordsuccI2: !x:set.x iIn ordsucc x const SNo_ : set set prop const SNoS_ : set set axiom SNoS_I: !x:set.ordinal x -> !y:set.!z:set.z iIn x -> SNo_ z y -> y iIn SNoS_ x axiom SNo_pos_eps_Lt: !x:set.nat_p x -> !y:set.y iIn SNoS_ (ordsucc x) -> Empty < y -> eps_ x < y axiom SNoLt_tra: !x:set.!y:set.!z:set.SNo x -> SNo y -> SNo z -> x < y -> y < z -> x < z axiom SNoLt_irref: !x:set.~ x < x const SNoLev : set set 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 add_SNo : set set set term + = add_SNo infix + 2281 2280 const minus_SNo : set set term - = minus_SNo lemma !x:set.!y:set.(!z:set.z iIn omega -> abs_SNo (y + - x) < eps_ z) -> Empty < y + - x -> SNoLev (y + - x) iIn omega -> abs_SNo (y + - x) = y + - x -> (y + - x) < eps_ (SNoLev (y + - x)) var x:set var y:set hyp x iIn SNoS_ omega hyp y iIn SNoS_ omega hyp !z:set.z iIn omega -> abs_SNo (y + - x) < eps_ z hyp Empty < y + - x claim ~ y + - x iIn SNoS_ omega