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