theorem extracted_formal_statement_37 {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)) →
              normA * ‖x - y‖ * normA * ‖x - y‖ ≤ ε * ‖x - y‖ → normA * ‖x - y‖ * normA ≤ ε := 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)
⊢ normA * ‖x - y‖ * normA * ‖x - y‖ ≤ ε * ‖x - y‖

Tactic: apply mul_le_mul_of_nonneg_right

State after:
[case h
 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)
 ⊢ normA * ‖x - y‖ * normA ≤ ε,
 case a0
 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)
 ⊢ 0 ≤ ‖x - y‖]

Relevant lemmas:
mul_le_mul_of_nonneg_right : ∀ {α : Type u_3} {a b c : α} [inst : Mul α] [inst_1 : Zero α] [inst_2 : Preorder α]
  [inst_3 : MulPosMono α], b ≤ c → 0 ≤ a → b * a ≤ c * a