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 SNo : set prop const SNoLt : set set prop term < = SNoLt infix < 2020 2020 const Empty : set const SNoS_ : set set const omega : 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 axiom SNo_prereal_incr_lower_pos: !x:set.SNo x -> Empty < 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 omega -> !P:prop.(!z:set.z iIn SNoS_ omega -> Empty < z -> z < x -> x < z + eps_ y -> P) -> P const real : set const SNoLev : set set const ordsucc : 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 axiom dneg: !P:prop.~ ~ P -> P const mul_SNo : set set set term * = mul_SNo infix * 2291 2290 lemma !x:set.!y:set.Empty < x -> Empty < y -> ~ x * y iIn real -> SNo x -> SNoLev x iIn ordsucc omega -> 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) -> (!z:set.z iIn omega -> ?w:set.w iIn SNoS_ omega & (w < x & x < w + eps_ z)) -> 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 -> ?w:set.w iIn SNoS_ omega & (w < y & y < w + eps_ z)) -> ~ !z:set.z iIn omega -> !P:prop.(!w:set.w iIn SNoS_ omega -> Empty < w -> w < x -> x < w + eps_ z -> P) -> P claim !x:set.x iIn real -> !y:set.y iIn real -> Empty < x -> Empty < y -> x * y iIn real