theorem extracted_formal_statement_1.{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) (s : α) (ε : ℝ) (εpos : ε > 0) (h : 0 < ‖A‖)
  (x : α) (hx : ‖s - x‖ ≤ ε / ‖A‖ ^ 2) (hzero : 0 < ‖A‖ ^ 2) :
  let t := x - s;
  s + t = x →
    ⟪A x, A x⟫_ℝ - ⟪A s, A s⟫_ℝ - ⟪2 • (ContinuousLinearMap.adjoint A) (A s), x - s⟫_ℝ = ⟪A (x - s), A (x - s)⟫_ℝ →
      |‖A (x - s)‖ ^ 2| ≤ ε * ‖s - x‖ := sorry

State before:
case h
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
s : α
ε : ℝ
εpos : ε > 0
h : 0 < ‖A‖
x : α
hx : ‖s - x‖ ≤ ε / ‖A‖ ^ 2
hzero : 0 < ‖A‖ ^ 2
t : α := x - s
t1 : s + t = x
this : ⟪A x, A x⟫_ℝ - ⟪A s, A s⟫_ℝ - ⟪2 • (ContinuousLinearMap.adjoint A) (A s), x - s⟫_ℝ = ⟪A (x - s), A (x - s)⟫_ℝ
⊢ |‖A (x - s)‖ ^ 2| ≤ ε * ‖s - x‖

Tactic: simp only [abs_pow, abs_norm, ge_iff_le]
    calc
  _ = ‖A (s - x)‖ ^ 2 := by rw [norm_comm]
  _ ≤ (‖A‖ * ‖s - x‖) ^ 2 := by
    rw [sq, sq, ← mul_self_le_mul_self_iff]
    apply ContinuousLinearMap.le_opNorm
    apply norm_nonneg
    simp [h, norm_nonneg (s - x)]
  _ = ‖A‖ ^ 2 * ‖s - x‖ ^ 2 := by linarith
    rcases(le_iff_eq_or_lt.1 $ norm_nonneg (s - x)) with h1 | _
    ·
  rw [← h1]
  simp only [ne_eq, OfNat.ofNat_ne_zero, not_false_eq_true, zero_pow, mul_zero, le_refl]
    ·
  calc
    _ = ‖A‖ ^ 2 * ‖s - x‖ * ‖s - x‖ := by nth_rw 2 [sq]; rw [mul_assoc]
    _ ≤ ‖A‖ ^ 2 * ‖s - x‖ * (ε / ‖A‖ ^ 2) :=
      by
      have : 0 ≤ ‖A‖ ^ 2 * ‖s - x‖ := by simp [hzero, norm_nonneg (s - x)]
      apply mul_le_mul_of_nonneg_left hx this
    _ = _ := by
      field_simp [hzero]
      ring_nf

State after:

Relevant lemmas:
error