theorem extracted_formal_statement_7 {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)) →
              ‖A *ᵥ (y - x)‖ ^ 2 ≤ (normA * ‖x - y‖) ^ 2 := 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)
aux2 : ∀ (u v : Fin ↑m → ℝ), u ⬝ᵥ u + (v ⬝ᵥ v - v ⬝ᵥ u * 2) = (u - v) ⬝ᵥ (u - v)
⊢ ‖A *ᵥ (y - x)‖ ^ 2 ≤ (normA * ‖x - y‖) ^ 2

Tactic: rw [norm_sub_rev]
    apply sq_le_sq' _ (norm_mul (y - x))
    calc
  -(normA * ‖y - x‖) ≤ 0 := by to_theorem simp;apply mul_nonneg;linarith [normApos]; apply norm_nonneg
  _ ≤ ‖A *ᵥ (y - x)‖₂ := by apply norm_nonneg

State after:

Relevant lemmas:
error