const Pi : set (set set) set term setexp = \x:set.\y:set.Pi y \z:set.x const SNo : set prop const In : set set prop term iIn = In infix iIn 2000 2000 const SNoS_ : set set const omega : set const SNoLt : set set prop term < = SNoLt infix < 2020 2020 const abs_SNo : set set const add_SNo : set set set term + = add_SNo infix + 2281 2280 const minus_SNo : set set term - = minus_SNo 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 axiom minus_SNo_prereal_2: !x:set.SNo x -> (!y:set.y iIn omega -> ?z:set.z iIn SNoS_ omega & (z < x & x < z + eps_ y)) -> !y:set.y iIn omega -> ?z:set.z iIn SNoS_ omega & (z < - x & - x < z + eps_ y) const ordinal : set prop axiom omega_ordinal: ordinal omega const ap : set set set axiom ap_Pi: !x:set.!f:set set.!y:set.!z:set.y iIn Pi x f -> z iIn x -> ap y z iIn f z 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 axiom SNo_prereal_incr_lower_approx: !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 omega -> ?z:set.z iIn SNoS_ omega & (z < x & x < z + eps_ y)) -> ?y:set.y iIn setexp (SNoS_ omega) omega & !z:set.z iIn omega -> ap y z < x & x < ap y z + eps_ z & !w:set.w iIn z -> ap y w < ap y z lemma !x:set.!y:set.SNo x -> y iIn setexp (SNoS_ omega) omega -> (!z:set.z iIn omega -> ap y z < - x & - x < ap y z + eps_ z & !w:set.w iIn z -> ap y w < ap y z) -> (!z:set.z iIn omega -> SNo (ap y z)) -> ?z:set.z iIn setexp (SNoS_ omega) omega & !w:set.w iIn omega -> (ap z w + - eps_ w) < x & x < ap z w & !u:set.u iIn w -> ap z w < ap z u var x:set hyp SNo x hyp !y:set.y iIn SNoS_ omega -> (!z:set.z iIn omega -> abs_SNo (y + - x) < eps_ z) -> y = x hyp !y:set.y iIn omega -> ?z:set.z iIn SNoS_ omega & (z < x & x < z + eps_ y) claim SNo - x -> ?y:set.y iIn setexp (SNoS_ omega) omega & !z:set.z iIn omega -> (ap y z + - eps_ z) < x & x < ap y z & !w:set.w iIn z -> ap y z < ap y w