theorem extracted_formal_statement_6.{u_5, u_4} {α : Type u_4} {β : Type u_5} [inst : NormedAddCommGroup α]
  [inst_1 : NormedAddCommGroup β] [inst_2 : InnerProductSpace ℝ α] [inst_3 : InnerProductSpace ℝ β]
  [inst_4 : CompleteSpace α] [inst_5 : CompleteSpace β] (A : α →L[ℝ] β) (c : β) (x : α) (ε : ℝ) (εpos : ε > 0) (x1 : α)
  (a : ‖x - x1‖ ≤ ε) : ‖⟪c, A x1 - A x - A (x1 - x)⟫_ℝ‖ ≤ ε * ‖x - x1‖ := sorry

State before:
case right
α : Type u_4
β : Type u_5
inst✝⁵ : NormedAddCommGroup α
inst✝⁴ : NormedAddCommGroup β
inst✝³ : InnerProductSpace ℝ α
inst✝² : InnerProductSpace ℝ β
inst✝¹ : CompleteSpace α
inst✝ : CompleteSpace β
A : α →L[ℝ] β
c : β
x : α
ε : ℝ
εpos : ε > 0
x1 : α
a✝ : ‖x - x1‖ ≤ ε
⊢ ‖⟪c, A x1 - A x - A (x1 - x)⟫_ℝ‖ ≤ ε * ‖x - x1‖

Tactic: simp only [map_sub, sub_self, inner_zero_right, norm_zero]
    have := norm_nonneg (x - x1)
    rwa [mul_nonneg_iff_right_nonneg_of_pos εpos]

State after:

Relevant lemmas:
sub_self : ∀ {G : Type u_1} [inst : AddGroup G] (a : G), a - a = 0
mul_nonneg_iff_right_nonneg_of_pos : ∀ {α : Type u} [inst : Semiring α] [inst_6 : LinearOrder α] {a b : α}
  [inst_7 : PosMulStrictMono α], 0 < a → (0 ≤ a * b ↔ 0 ≤ b)
norm_nonneg : ∀ {E : Type u_5} [inst : SeminormedAddGroup E] (a : E), 0 ≤ ‖a‖
inner_zero_right : ∀ {𝕜 : Type u_1} {E : Type u_2} [inst : RCLike 𝕜] [inst_6 : SeminormedAddCommGroup E]
  [inst_7 : InnerProductSpace 𝕜 E] (x : E), ⟪x, 0⟫_𝕜 = 0
norm_zero : ∀ {E : Type u_5} [inst : SeminormedAddGroup E], ‖0‖ = 0
map_sub : ∀ {G : Type u_7} {H : Type u_8} {F : Type u_9} [inst : FunLike F G H] [inst_6 : AddGroup G]
  [inst_7 : SubtractionMonoid H] [inst_8 : AddMonoidHomClass F G H] (f : F) (a b : G), f (a - b) = f a - f b