∀ {ι : Type u_1} {R : ι → Type u_2} {A : (i : ι) → Set (R i)} [inst : (i : ι) → TopologicalSpace (R i)],
  (∀ (i : ι), IsOpen (A i)) →
    ∀ {X : Type u_3} {Y : Type u_4} [inst_1 : TopologicalSpace X] [inst_2 : TopologicalSpace Y]
      {f : Y × RestrictedProduct (fun i => R i) (fun i => A i) Filter.cofinite → X},
      Continuous f ↔
        ∀ (S : Set ι) (hS : Filter.cofinite ≤ Filter.principal S),
          Continuous (f ∘ Prod.map id (RestrictedProduct.inclusion R A hS))