∀ {R : Type u_1} [inst : CommRing R] {A : Type u_5} {B : Type u_6} [inst_1 : Ring A] [inst_2 : Algebra R A]
  [inst_3 : Ring B] [inst_4 : Algebra R B] [Algebra.IsIntegral R A] [Algebra.IsIntegral R B],
  Algebra.IsIntegral R (A × B)