∀ (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 →+* Ring.Perfection S p) (n : ℕ),
  ((Perfection.coeff S p 0).comp f) ((⇑↑(frobeniusEquiv R p).symm)^[n + 1] 1) ^ p =
    ((Perfection.coeff S p 0).comp f) ((⇑↑(frobeniusEquiv R p).symm)^[n] 1)