theorem extracted_formal_statement_1.{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] (x : X) (h : (ContinuousLinearMap.adjoint A) (A x) = 0) :
  ⟪x, (ContinuousLinearMap.adjoint A) (A x)⟫_ℝ = 0 := sorry

State before:
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
x : X
h : (ContinuousLinearMap.adjoint A) (A x) = 0
⊢ ⟪x, (ContinuousLinearMap.adjoint A) (A x)⟫_ℝ = 0

Tactic: rw [h, inner_zero_right]

State after:

Relevant lemmas:
inner_zero_right : ∀ {𝕜 : Type u_1} {E : Type u_2} [inst : RCLike 𝕜] [inst_6 : SeminormedAddCommGroup E]
  [inst_7 : InnerProductSpace 𝕜 E] (x : E), ⟪x, 0⟫_𝕜 = 0