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 term nIn = \x:set.\y:set.~ x iIn y term TransSet = \x:set.!y:set.y iIn x -> Subq y x const Pi : set (set set) set term setexp = \x:set.\y:set.Pi y \z:set.x const ordinal : set prop const omega : set axiom omega_ordinal: ordinal omega axiom omega_TransSet: TransSet omega const ordsucc : set set axiom ordsuccE: !x:set.!y:set.y iIn ordsucc x -> y iIn x | y = x const SNo : set prop const SNo_ : set set prop const SNoLev : set set axiom SNoLev_: !x:set.SNo x -> SNo_ (SNoLev x) x 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 const SNoL : set set const SNoLt : set set prop term < = SNoLt infix < 2020 2020 axiom SNoL_E: !x:set.SNo x -> !y:set.y iIn SNoL x -> !P:prop.(SNo y -> SNoLev y iIn SNoLev x -> y < x -> P) -> P const Empty : set const mul_SNo : set set set term * = mul_SNo infix * 2291 2290 const real : set 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 const SNoR : set set lemma !x:set.!y:set.Empty < x -> Empty < y -> ~ x * y iIn real -> SNo x -> x iIn SNoS_ (ordsucc omega) -> x < omega -> (!z:set.z iIn SNoS_ omega -> (!w:set.w iIn omega -> abs_SNo (z + - x) < eps_ w) -> z = x) -> SNo y -> SNoLev y iIn ordsucc omega -> y iIn SNoS_ (ordsucc omega) -> y < omega -> (!z:set.z iIn SNoS_ omega -> (!w:set.w iIn omega -> abs_SNo (z + - y) < eps_ w) -> z = y) -> (!z:set.z iIn omega -> !P:prop.(!w:set.w iIn SNoS_ omega -> Empty < w -> w < x -> x < w + eps_ z -> P) -> P) -> (!z:set.z iIn omega -> !P:prop.(!w:set.w iIn SNoS_ omega -> Empty < w -> w < y -> y < w + eps_ z -> P) -> P) -> SNo (x * y) -> SNo - x * y -> (!z:set.SNo z -> SNoLev z iIn omega -> SNoLev z iIn SNoLev (x * y)) -> Subq (SNoL x) (SNoS_ omega) -> Subq (SNoR x) (SNoS_ omega) -> ~ Subq (SNoL y) (SNoS_ omega) var x:set var y:set hyp Empty < x hyp Empty < y hyp ~ x * y iIn real hyp SNo x hyp SNoLev x iIn ordsucc omega hyp x iIn SNoS_ (ordsucc omega) hyp x < omega hyp !z:set.z iIn SNoS_ omega -> (!w:set.w iIn omega -> abs_SNo (z + - x) < eps_ w) -> z = x hyp SNo y hyp SNoLev y iIn ordsucc omega hyp y iIn SNoS_ (ordsucc omega) hyp y < omega hyp !z:set.z iIn SNoS_ omega -> (!w:set.w iIn omega -> abs_SNo (z + - y) < eps_ w) -> z = y hyp !z:set.z iIn omega -> !P:prop.(!w:set.w iIn SNoS_ omega -> Empty < w -> w < x -> x < w + eps_ z -> P) -> P hyp !z:set.z iIn omega -> !P:prop.(!w:set.w iIn SNoS_ omega -> Empty < w -> w < y -> y < w + eps_ z -> P) -> P hyp SNo (x * y) hyp SNo - x * y hyp !z:set.SNo z -> SNoLev z iIn omega -> SNoLev z iIn SNoLev (x * y) hyp Subq (SNoL x) (SNoS_ omega) claim ~ Subq (SNoR x) (SNoS_ omega)