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 SNoLt : set set prop term < = SNoLt infix < 2020 2020 axiom SNoLt_irref: !x:set.~ x < x axiom FalseE: ~ False const SNo : set prop const Empty : set const add_SNo : set set set term + = add_SNo infix + 2281 2280 const minus_SNo : set set term - = minus_SNo axiom SNoLt_minus_pos: !x:set.!y:set.SNo x -> SNo y -> x < y -> Empty < y + - x axiom SNo_minus_SNo: !x:set.SNo x -> SNo - x axiom SNo_add_SNo: !x:set.!y:set.SNo x -> SNo y -> SNo (x + y) const omega : set const eps_ : set set axiom SNo_eps_: !x:set.x iIn omega -> SNo (eps_ x) 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 dneg: !P:prop.~ ~ P -> P axiom xm: !P:prop.P | ~ P const SNoR : set set const SNoLev : set set axiom SNoR_E: !x:set.SNo x -> !y:set.y iIn SNoR x -> !P:prop.(SNo y -> SNoLev y iIn SNoLev x -> x < y -> P) -> P const mul_SNo : set set set term * = mul_SNo infix * 2291 2290 const real : set const SNoS_ : set set const ordsucc : set set const SNoL : 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 -> 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 (SNoR x) (SNoS_ omega) -> Subq (SNoL y) (SNoS_ omega) -> Subq (SNoR y) (SNoS_ omega) -> (!z:set.z iIn SNoL x -> !P:prop.(!w:set.w iIn omega -> eps_ w <= x + - z -> P) -> P) -> ~ !z:set.z iIn SNoR x -> !P:prop.(!w:set.w iIn omega -> eps_ w <= z + - x -> P) -> P var x:set var y:set hyp Empty < x hyp Empty < y hyp ~ x * y iIn real hyp SNo x 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 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) hyp Subq (SNoR x) (SNoS_ omega) hyp Subq (SNoL y) (SNoS_ omega) hyp Subq (SNoR y) (SNoS_ omega) claim ~ !z:set.z iIn SNoL x -> !P:prop.(!w:set.w iIn omega -> eps_ w <= x + - z -> P) -> P