theorem extracted_formal_statement_12 {n m : ℕ+} {A : Matrix (Fin ↑m) (Fin ↑n) ℝ} (hA : ¬A = 0)
  (x : EuclideanSpace ℝ (Fin ↑n)) (h_original_goal : HasGradientAt (fun x => A *ᵥ x ⬝ᵥ A *ᵥ x) (2 • Aᵀ *ᵥ A *ᵥ x) x)
  (ε : ℝ) :
  ε > 0 →
    ∃ δ > 0,
      ∀ (x' : EuclideanSpace ℝ (Fin ↑n)),
        ‖x - x'‖ ≤ δ →
          ‖A *ᵥ x' ⬝ᵥ A *ᵥ x' - A *ᵥ x ⬝ᵥ A *ᵥ x - inner (2 • Aᵀ *ᵥ A *ᵥ x) (x' - x)‖ ≤ ε * ‖x - x'‖ := sorry

State before:
case neg
n m : ℕ+
A : Matrix (Fin ↑m) (Fin ↑n) ℝ
hA : ¬A = 0
x : EuclideanSpace ℝ (Fin ↑n)
⊢ HasGradientAt (fun x => A *ᵥ x ⬝ᵥ A *ᵥ x) (2 • Aᵀ *ᵥ A *ᵥ x) x

Tactic: rw [HasGradient_iff_Convergence_Point]

State after:
[case neg
 n m : ℕ+
 A : Matrix (Fin ↑m) (Fin ↑n) ℝ
 hA : ¬A = 0
 x : EuclideanSpace ℝ (Fin ↑n)
 ⊢ ∀ ε > 0,
     ∃ δ > 0,
       ∀ (x' : EuclideanSpace ℝ (Fin ↑n)),
         ‖x - x'‖ ≤ δ → ‖A *ᵥ x' ⬝ᵥ A *ᵥ x' - A *ᵥ x ⬝ᵥ A *ᵥ x - inner (2 • Aᵀ *ᵥ A *ᵥ x) (x' - x)‖ ≤ ε * ‖x - x'‖]

Relevant lemmas:
HasGradient_iff_Convergence_Point : ∀ {E : Type u_1} [inst : NormedAddCommGroup E] [inst_1 : InnerProductSpace ℝ E]
  [inst_2 : CompleteSpace E] {x : E} {f : E → ℝ} {f'x : E},
  HasGradientAt f f'x x ↔ ∀ ε > 0, ∃ δ > 0, ∀ (x' : E), ‖x - x'‖ ≤ δ → ‖f x' - f x - inner f'x (x' - x)‖ ≤ ε * ‖x - x'‖