∀ {R : Type u_2} {S : Type u_1} [inst : CommRing R] [inst_1 : CommRing S] [inst_2 : Algebra R S] {x : S},
  IsIntegral R x →
    ∀ {i : ℕ},
      i ≤ (minpolyDiv R x).natDegree →
        (minpolyDiv R x).coeff ((minpolyDiv R x).natDegree - i) - x ^ i ∈
          Submodule.span R ((fun x_1 => x ^ x_1) '' Set.Iio i)