theorem extracted_formal_statement_24 {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 => y x_1 - x x_1) ⬝ᵥ fun x_1 => ((Aᵀ * A) *ᵥ x) x_1) = (Aᵀ * A) *ᵥ x ⬝ᵥ (y - x) →
            ∀ (u v : Fin ↑m → ℝ),
              u ⬝ᵥ u + (v ⬝ᵥ v - v ⬝ᵥ u * 2) = (u - v) ⬝ᵥ (u - v) →
                u ⬝ᵥ u + (v ⬝ᵥ v - v ⬝ᵥ u * 2) = u ⬝ᵥ u - (v ⬝ᵥ u + v ⬝ᵥ u) + v ⬝ᵥ v := 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
aux1 : ((fun x_1 => y x_1 - x x_1) ⬝ᵥ fun x_1 => ((Aᵀ * A) *ᵥ x) x_1) = (Aᵀ * A) *ᵥ x ⬝ᵥ (y - x)
u v : Fin ↑m → ℝ
⊢ u ⬝ᵥ u + (v ⬝ᵥ v - v ⬝ᵥ u * 2) = (u - v) ⬝ᵥ (u - v)

Tactic: rw [dotProduct_sub, sub_dotProduct, sub_dotProduct, ← sub_add, sub_sub, dotProduct_comm u v]

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
 aux1 : ((fun x_1 => y x_1 - x x_1) ⬝ᵥ fun x_1 => ((Aᵀ * A) *ᵥ x) x_1) = (Aᵀ * A) *ᵥ x ⬝ᵥ (y - x)
 u v : Fin ↑m → ℝ
 ⊢ u ⬝ᵥ u + (v ⬝ᵥ v - v ⬝ᵥ u * 2) = u ⬝ᵥ u - (v ⬝ᵥ u + v ⬝ᵥ u) + v ⬝ᵥ v]

Relevant lemmas:
dotProduct_comm : ∀ {m : Type u_2} {α : Type v} [inst : Fintype m] [inst_1 : AddCommMonoid α] [inst_2 : CommMagma α]
  (v w : m → α), v ⬝ᵥ w = w ⬝ᵥ v
dotProduct_sub : ∀ {m : Type u_2} {α : Type v} [inst : Fintype m] [inst_1 : NonUnitalNonAssocRing α] (u v w : m → α),
  u ⬝ᵥ (v - w) = u ⬝ᵥ v - u ⬝ᵥ w
sub_add : ∀ {α : Type u_1} [inst : SubtractionCommMonoid α] (a b c : α), a - b + c = a - (b - c)
sub_sub : ∀ {α : Type u_1} [inst : SubtractionCommMonoid α] (a b c : α), a - b - c = a - (b + c)
sub_dotProduct : ∀ {m : Type u_2} {α : Type v} [inst : Fintype m] [inst_1 : NonUnitalNonAssocRing α] (u v w : m → α),
  (u - v) ⬝ᵥ w = u ⬝ᵥ w - v ⬝ᵥ w