∀ {X : AlgebraicGeometry.Scheme} {ι : Sort u_1} [Finite ι] {S : ι → ↑X.affineOpens},
  ⨆ i, ↑(S i) = ⊤ →
    (AlgebraicGeometry.IsNoetherian X ↔ ∀ (i : ι), IsNoetherianRing ↑(X.presheaf.obj (Opposite.op ↑(S i))))