∀ {K : Type u_2} {S : Type u_1} [inst : Field K] [inst_1 : CommRing S] [inst_2 : Algebra K S] {x : S},
  Function.Injective ⇑(algebraMap (↥(Algebra.adjoin K {x})) S) →
    IsIntegral K ⟨x, ⋯⟩ →
      ∀ ⦃x_1 : ↥(Algebra.adjoin K {x})⦄, x_1 ∈ ⊤ → x_1 ∈ Submodule.span K (Set.range fun i => ⟨x, ⋯⟩ ^ ↑i)