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 mul_SNo : set set set term * = mul_SNo infix * 2291 2290 axiom SNo_mul_SNo: !x:set.!y:set.SNo x -> SNo y -> SNo (x * y) const SNoR : set set const omega : set const SNoLe : set set prop term <= = SNoLe infix <= 2020 2020 const eps_ : set set const add_SNo : set set set term + = add_SNo infix + 2281 2280 const minus_SNo : set set term - = minus_SNo const SNoLt : set set prop term < = SNoLt infix < 2020 2020 const abs_SNo : set set lemma !x:set.!y:set.!z:set.!w:set.!u:set.SNo x -> SNo y -> SNo (x * y) -> (!v:set.v iIn SNoR x -> !P:prop.(!x2:set.x2 iIn omega -> eps_ x2 <= v + - x -> P) -> P) -> (!v:set.v iIn SNoR y -> !P:prop.(!x2:set.x2 iIn omega -> eps_ x2 <= v + - y -> P) -> P) -> (!v:set.v iIn omega -> abs_SNo (z + - x * y) < eps_ v) -> SNo z -> z < x * y -> w iIn SNoR x -> u iIn SNoR y -> SNo w -> SNo u -> (z + w * u) <= w * y + x * u -> SNo (w * u) -> SNo (x * u) -> SNo - x * u -> ~ SNo (w * y) var x:set var y:set var z:set var w:set var u:set hyp SNo x hyp SNo y hyp SNo (x * y) hyp !v:set.v iIn SNoR x -> !P:prop.(!x2:set.x2 iIn omega -> eps_ x2 <= v + - x -> P) -> P hyp !v:set.v iIn SNoR y -> !P:prop.(!x2:set.x2 iIn omega -> eps_ x2 <= v + - y -> P) -> P hyp !v:set.v iIn omega -> abs_SNo (z + - x * y) < eps_ v hyp SNo z hyp z < x * y hyp w iIn SNoR x hyp u iIn SNoR y hyp SNo w hyp SNo u hyp (z + w * u) <= w * y + x * u hyp SNo (w * u) hyp SNo (x * u) claim ~ SNo - x * u