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 omega : set const add_SNo : set set set term + = add_SNo infix + 2281 2280 axiom add_SNo_In_omega: !x:set.x iIn omega -> !y:set.y iIn omega -> x + y iIn omega const SNo : set prop const mul_SNo : set set set term * = mul_SNo infix * 2291 2290 const SNoLt : set set prop term < = SNoLt infix < 2020 2020 const abs_SNo : set set const minus_SNo : set set term - = minus_SNo const eps_ : set set const SNoLe : set set prop term <= = SNoLe infix <= 2020 2020 lemma !x:set.!y:set.!z:set.!w:set.!u:set.!v:set.!x2:set.SNo x -> SNo y -> SNo (x * y) -> (!y2:set.y2 iIn omega -> abs_SNo (z + - x * y) < eps_ y2) -> SNo z -> z < x * y -> SNo w -> SNo u -> (z + w * u) <= w * y + x * u -> SNo (w * u) -> SNo (x * u) -> SNo - x * u -> SNo (w * y) -> SNo - w * y -> SNo (w + - x) -> SNo (u + - y) -> v iIn omega -> eps_ v <= w + - x -> x2 iIn omega -> eps_ x2 <= u + - y -> SNo (eps_ v) -> SNo (eps_ x2) -> SNo (eps_ (v + x2)) -> SNo (eps_ v * eps_ x2) -> ~ abs_SNo (z + - x * y) < eps_ (v + x2) var x:set var y:set var z:set var w:set var u:set var v:set var x2:set hyp SNo x hyp SNo y hyp SNo (x * y) hyp !y2:set.y2 iIn omega -> abs_SNo (z + - x * y) < eps_ y2 hyp SNo z hyp z < x * y hyp SNo w hyp SNo u hyp (z + w * u) <= w * y + x * u hyp SNo (w * u) hyp SNo (x * u) hyp SNo - x * u hyp SNo (w * y) hyp SNo - w * y hyp SNo (w + - x) hyp SNo (u + - y) hyp v iIn omega hyp eps_ v <= w + - x hyp x2 iIn omega hyp eps_ x2 <= u + - y hyp SNo (eps_ v) hyp SNo (eps_ x2) hyp SNo (eps_ (v + x2)) claim ~ SNo (eps_ v * eps_ x2)