theorem extracted_formal_statement_6.{u_2, u_1} {X : Type u_1} {Y : Type u_2} [inst : NormedAddCommGroup X]
  [inst_1 : InnerProductSpace ℝ X] [inst_2 : NormedAddCommGroup Y] [inst_3 : InnerProductSpace ℝ Y] (A : X →L[ℝ] Y)
  [inst_4 : CompleteSpace X] [inst_5 : CompleteSpace Y] (this : IsBoundedLinearMap ℝ ⇑(ContinuousLinearMap.adjoint A))
  (M : ℝ) (h1 : 0 < M ∧ ∀ (x : Y), ‖(ContinuousLinearMap.adjoint A) x‖ ≤ M * ‖x‖) (x : X)
  (h2 : ‖(ContinuousLinearMap.adjoint A) (A x)‖ ≤ M * ‖A x‖) :
  ‖((ContinuousLinearMap.adjoint A).comp A) x‖ ≤ M * ‖A x‖ := sorry

State before:
case right
X : Type u_1
Y : Type u_2
inst✝⁵ : NormedAddCommGroup X
inst✝⁴ : InnerProductSpace ℝ X
inst✝³ : NormedAddCommGroup Y
inst✝² : InnerProductSpace ℝ Y
A : X →L[ℝ] Y
inst✝¹ : CompleteSpace X
inst✝ : CompleteSpace Y
this : IsBoundedLinearMap ℝ ⇑(ContinuousLinearMap.adjoint A)
M : ℝ
h1 : 0 < M ∧ ∀ (x : Y), ‖(ContinuousLinearMap.adjoint A) x‖ ≤ M * ‖x‖
x : X
h2 : ‖(ContinuousLinearMap.adjoint A) (A x)‖ ≤ M * ‖A x‖
⊢ ‖((ContinuousLinearMap.adjoint A).comp A) x‖ ≤ M * ‖A x‖

Tactic: exact h2

State after:

Relevant lemmas: