theorem extracted_formal_statement_21 {n m : ℕ+} {A : Matrix (Fin ↑m) (Fin ↑n) ℝ} (hA : ¬A = 0)
  (x : EuclideanSpace ℝ (Fin ↑n)) (ε : ℝ) (εpos : ε > 0) :
  let normA := ‖(toEuclideanLin ≪≫ₗ LinearMap.toContinuousLinearMap) A‖;
  (∀ (x : EuclideanSpace ℝ (Fin ↑n)), ‖A *ᵥ x‖ ≤ normA * ‖x‖) →
    0 < normA →
      ∀ (y : EuclideanSpace ℝ (Fin ↑n)),
        ‖x - y‖ ≤ ε / normA ^ 2 →
          ((fun x_1 => ((Aᵀ * A) *ᵥ x) x_1) ⬝ᵥ fun x_1 => y x_1 - x x_1) = (Aᵀ * A) *ᵥ x ⬝ᵥ (y - x) →
            ∑ i, ((Aᵀ * A) *ᵥ x) i * (y i - x i) = ∑ i, ((Aᵀ * A) *ᵥ x) i * (y - x) i := sorry

State before:
n m : ℕ+
A : Matrix (Fin ↑m) (Fin ↑n) ℝ
hA : ¬A = 0
x : EuclideanSpace ℝ (Fin ↑n)
ε : ℝ
εpos : ε > 0
normA : ℝ := ‖(toEuclideanLin ≪≫ₗ LinearMap.toContinuousLinearMap) A‖
norm_mul : ∀ (x : EuclideanSpace ℝ (Fin ↑n)), ‖A *ᵥ x‖ ≤ normA * ‖x‖
normApos : 0 < normA
y : EuclideanSpace ℝ (Fin ↑n)
ydist : ‖x - y‖ ≤ ε / normA ^ 2
⊢ ((fun x_1 => ((Aᵀ * A) *ᵥ x) x_1) ⬝ᵥ fun x_1 => y x_1 - x x_1) = (Aᵀ * A) *ᵥ x ⬝ᵥ (y - x)

Tactic: rw [dotProduct, dotProduct]

State after:
[n m : ℕ+
 A : Matrix (Fin ↑m) (Fin ↑n) ℝ
 hA : ¬A = 0
 x : EuclideanSpace ℝ (Fin ↑n)
 ε : ℝ
 εpos : ε > 0
 normA : ℝ := ‖(toEuclideanLin ≪≫ₗ LinearMap.toContinuousLinearMap) A‖
 norm_mul : ∀ (x : EuclideanSpace ℝ (Fin ↑n)), ‖A *ᵥ x‖ ≤ normA * ‖x‖
 normApos : 0 < normA
 y : EuclideanSpace ℝ (Fin ↑n)
 ydist : ‖x - y‖ ≤ ε / normA ^ 2
 ⊢ ∑ i, ((Aᵀ * A) *ᵥ x) i * (y i - x i) = ∑ i, ((Aᵀ * A) *ᵥ x) i * (y - x) i]

Relevant lemmas: