∀ (R : Type u_2) [inst : CommRing R] (K : Type u_1) [inst_1 : Field K] [inst_2 : Algebra (Polynomial R) K]
  {ι : Type u_3} {g : ι → Polynomial R} (a : ι) (b : Finset ι),
  a ∉ b →
    ∀ (r₁ : Polynomial R) (r : ι → Polynomial R),
      ∀ x ∈ b,
        (algebraMap (Polynomial R) K) (r x) / (algebraMap (Polynomial R) K) (g x) =
          (algebraMap (Polynomial R) K) (if x = a then r₁ else r x) / (algebraMap (Polynomial R) K) (g x)