∀ {k : Type u_3} {K : Type u_4} [inst : Field k] [inst_1 : Field K] [inst_2 : Algebra k K] [inst_3 : IsGalois k K]
  (g g_1 : ↑(ProfiniteGrp.limit (InfiniteGalois.asProfiniteGaloisGroupFunctor k K)).toProfinite.toTop),
  g = g_1 → ∀ (a a_1 : K), a = a_1 → InfiniteGalois.toAlgEquivAux✝ g a = InfiniteGalois.toAlgEquivAux✝¹ g_1 a_1