∀ {𝕜 : Type u_3} {𝕜₂ : Type u_4} {F : Type u_1} [inst : NormedAddCommGroup F] [inst_1 : NontriviallyNormedField 𝕜]
  [inst_2 : NontriviallyNormedField 𝕜₂] [inst_3 : NormedSpace 𝕜₂ F] {σ₁₂ : 𝕜 →+* 𝕜₂} {E' : Type u_2}
  [inst_4 : SeminormedAddCommGroup E'] [inst_5 : NormedSpace 𝕜 E'] [inst_6 : RingHomIsometric σ₁₂] (f : E' → F)
  {s : Set (E' →SL[σ₁₂] F)},
  Bornology.IsBounded s →
    ∀ (hf : f ∈ closure (DFunLike.coe '' s)), ∃ C, ∀ (x : E'), ‖(linearMapOfMemClosureRangeCoe f ⋯) x‖ ≤ C * ‖x‖