∀ (p : ℕ) [hp : Fact (Nat.Prime p)] (R : Type u_2) [inst : CommSemiring R] [inst_1 : CharP R p]
  [inst_2 : PerfectRing R p] (S : Type u_1) [inst_3 : CommSemiring S] [inst_4 : CharP S p] (f : R →+* Perfection S p)
  (x x_1 : R) (n : ℕ),
  ((Perfection.coeff S p 0).comp f) ((⇑↑(frobeniusEquiv R p).symm)^[n + 1] (x * x_1)) ^ p =
    ((Perfection.coeff S p 0).comp f) ((⇑↑(frobeniusEquiv R p).symm)^[n] (x * x_1))