-- Module: Mathlib.Analysis.InnerProductSpace.OfNorm
Goal:
𝕜 : Type u_1
inst : RCLike 𝕜
E : Type u_2
inst_1 : NormedAddCommGroup E
inst_2 : NormedSpace 𝕜 E
inst_3 : InnerProductSpaceable E
x y z : E
h8 :
  HMul.hMul (α := 𝕜) (β := 𝕜) ((𝓚 : (a : ℝ) → 𝕜) ‖HSMul.hSMul.{u_1, u_2, u_2} (α := 𝕜) I y‖ : 𝕜)
        ((𝓚 : (a : ℝ) → 𝕜) ‖HSMul.hSMul.{u_1, u_2, u_2} (α := 𝕜) I y‖ : 𝕜) +
      HMul.hMul (α := 𝕜) (β := 𝕜)
        ((𝓚 : (a : ℝ) → 𝕜) ‖HSMul.hSMul (α := ℤ) (-2) z + HSMul.hSMul.{u_1, u_2, u_2} (α := 𝕜) I y‖ : 𝕜)
        ((𝓚 : (a : ℝ) → 𝕜) ‖HSMul.hSMul (α := ℤ) (-2) z + HSMul.hSMul.{u_1, u_2, u_2} (α := 𝕜) I y‖ : 𝕜) =
    2 *
      (HMul.hMul (α := 𝕜) (β := 𝕜)
          ((𝓚 : (a : ℝ) → 𝕜) ‖HSMul.hSMul (α := ℤ) (-1) z + HSMul.hSMul.{u_1, u_2, u_2} (α := 𝕜) I y‖ : 𝕜)
          ((𝓚 : (a : ℝ) → 𝕜) ‖HSMul.hSMul (α := ℤ) (-1) z + HSMul.hSMul.{u_1, u_2, u_2} (α := 𝕜) I y‖ : 𝕜) +
        HMul.hMul (α := 𝕜) (β := 𝕜) ((𝓚 : (a : ℝ) → 𝕜) ‖z‖ : 𝕜) ((𝓚 : (a : ℝ) → 𝕜) ‖z‖ : 𝕜))
h7 :
  HMul.hMul (α := 𝕜) (β := 𝕜) ((𝓚 : (a : ℝ) → 𝕜) ‖HSMul.hSMul.{u_1, u_2, u_2} (α := 𝕜) I y‖ : 𝕜)
        ((𝓚 : (a : ℝ) → 𝕜) ‖HSMul.hSMul.{u_1, u_2, u_2} (α := 𝕜) I y‖ : 𝕜) +
      HMul.hMul (α := 𝕜) (β := 𝕜)
        ((𝓚 : (a : ℝ) → 𝕜) ‖HSMul.hSMul (α := ℤ) 2 z + HSMul.hSMul.{u_1, u_2, u_2} (α := 𝕜) I y‖ : 𝕜)
        ((𝓚 : (a : ℝ) → 𝕜) ‖HSMul.hSMul (α := ℤ) 2 z + HSMul.hSMul.{u_1, u_2, u_2} (α := 𝕜) I y‖ : 𝕜) =
    2 *
      (HMul.hMul (α := 𝕜) (β := 𝕜) ((𝓚 : (a : ℝ) → 𝕜) ‖z‖ : 𝕜) ((𝓚 : (a : ℝ) → 𝕜) ‖z‖ : 𝕜) +
        HMul.hMul (α := 𝕜) (β := 𝕜) ((𝓚 : (a : ℝ) → 𝕜) ‖z + HSMul.hSMul.{u_1, u_2, u_2} (α := 𝕜) I y‖ : 𝕜)
          ((𝓚 : (a : ℝ) → 𝕜) ‖z + HSMul.hSMul.{u_1, u_2, u_2} (α := 𝕜) I y‖ : 𝕜))
h6 :
  HMul.hMul (α := 𝕜) (β := 𝕜)
        ((𝓚 : (a : ℝ) → 𝕜) ‖HSMul.hSMul (α := ℤ) (-2) z + HSMul.hSMul.{u_1, u_2, u_2} (α := 𝕜) I y‖ : 𝕜)
        ((𝓚 : (a : ℝ) → 𝕜) ‖HSMul.hSMul (α := ℤ) (-2) z + HSMul.hSMul.{u_1, u_2, u_2} (α := 𝕜) I y‖ : 𝕜) +
      HMul.hMul (α := 𝕜) (β := 𝕜)
        ((𝓚 : (a : ℝ) → 𝕜)
            ‖HSMul.hSMul (α := ℤ) 2 (HSMul.hSMul.{u_1, u_2, u_2} (α := 𝕜) I x) +
                HSMul.hSMul.{u_1, u_2, u_2} (α := 𝕜) I y‖ :
          𝕜)
        ((𝓚 : (a : ℝ) → 𝕜)
            ‖HSMul.hSMul (α := ℤ) 2 (HSMul.hSMul.{u_1, u_2, u_2} (α := 𝕜) I x) +
                HSMul.hSMul.{u_1, u_2, u_2} (α := 𝕜) I y‖ :
          𝕜) =
    2 *
      (HMul.hMul (α := 𝕜) (β := 𝕜)
          ((𝓚 : (a : ℝ) → 𝕜)
              ‖HSMul.hSMul (α := ℤ) (-1) z +
                  (HSMul.hSMul.{u_1, u_2, u_2} (α := 𝕜) I x + HSMul.hSMul.{u_1, u_2, u_2} (α := 𝕜) I y)‖ :
            𝕜)
          ((𝓚 : (a : ℝ) → 𝕜)
              ‖HSMul.hSMul (α := ℤ) (-1) z +
                  (HSMul.hSMul.{u_1, u_2, u_2} (α := 𝕜) I x + HSMul.hSMul.{u_1, u_2, u_2} (α := 𝕜) I y)‖ :
            𝕜) +
        HMul.hMul (α := 𝕜) (β := 𝕜) ((𝓚 : (a : ℝ) → 𝕜) ‖z + HSMul.hSMul.{u_1, u_2, u_2} (α := 𝕜) I x‖ : 𝕜)
          ((𝓚 : (a : ℝ) → 𝕜) ‖z + HSMul.hSMul.{u_1, u_2, u_2} (α := 𝕜) I x‖ : 𝕜))
h5 :
  HMul.hMul (α := 𝕜) (β := 𝕜)
        ((𝓚 : (a : ℝ) → 𝕜) ‖HSMul.hSMul (α := ℤ) 2 z + HSMul.hSMul.{u_1, u_2, u_2} (α := 𝕜) I y‖ : 𝕜)
        ((𝓚 : (a : ℝ) → 𝕜) ‖HSMul.hSMul (α := ℤ) 2 z + HSMul.hSMul.{u_1, u_2, u_2} (α := 𝕜) I y‖ : 𝕜) +
      HMul.hMul (α := 𝕜) (β := 𝕜)
        ((𝓚 : (a : ℝ) → 𝕜)
            ‖HSMul.hSMul (α := ℤ) 2 (HSMul.hSMul.{u_1, u_2, u_2} (α := 𝕜) I x) +
                HSMul.hSMul.{u_1, u_2, u_2} (α := 𝕜) I y‖ :
          𝕜)
        ((𝓚 : (a : ℝ) → 𝕜)
            ‖HSMul.hSMul (α := ℤ) 2 (HSMul.hSMul.{u_1, u_2, u_2} (α := 𝕜) I x) +
                HSMul.hSMul.{u_1, u_2, u_2} (α := 𝕜) I y‖ :
          𝕜) =
    2 *
      (HMul.hMul (α := 𝕜) (β := 𝕜)
          ((𝓚 : (a : ℝ) → 𝕜)
              ‖z + (HSMul.hSMul.{u_1, u_2, u_2} (α := 𝕜) I x + HSMul.hSMul.{u_1, u_2, u_2} (α := 𝕜) I y)‖ :
            𝕜)
          ((𝓚 : (a : ℝ) → 𝕜)
              ‖z + (HSMul.hSMul.{u_1, u_2, u_2} (α := 𝕜) I x + HSMul.hSMul.{u_1, u_2, u_2} (α := 𝕜) I y)‖ :
            𝕜) +
        HMul.hMul (α := 𝕜) (β := 𝕜)
          ((𝓚 : (a : ℝ) → 𝕜) ‖HSMul.hSMul (α := ℤ) (-1) z + HSMul.hSMul.{u_1, u_2, u_2} (α := 𝕜) I x‖ : 𝕜)
          ((𝓚 : (a : ℝ) → 𝕜) ‖HSMul.hSMul (α := ℤ) (-1) z + HSMul.hSMul.{u_1, u_2, u_2} (α := 𝕜) I x‖ : 𝕜))
h4 :
  HMul.hMul (α := 𝕜) (β := 𝕜) ((𝓚 : (a : ℝ) → 𝕜) ‖y‖ : 𝕜) ((𝓚 : (a : ℝ) → 𝕜) ‖y‖ : 𝕜) +
      HMul.hMul (α := 𝕜) (β := 𝕜) ((𝓚 : (a : ℝ) → 𝕜) ‖y + HSMul.hSMul (α := ℤ) (-2) z‖ : 𝕜)
        ((𝓚 : (a : ℝ) → 𝕜) ‖y + HSMul.hSMul (α := ℤ) (-2) z‖ : 𝕜) =
    2 *
      (HMul.hMul (α := 𝕜) (β := 𝕜) ((𝓚 : (a : ℝ) → 𝕜) ‖y + HSMul.hSMul (α := ℤ) (-1) z‖ : 𝕜)
          ((𝓚 : (a : ℝ) → 𝕜) ‖y + HSMul.hSMul (α := ℤ) (-1) z‖ : 𝕜) +
        HMul.hMul (α := 𝕜) (β := 𝕜) ((𝓚 : (a : ℝ) → 𝕜) ‖z‖ : 𝕜) ((𝓚 : (a : ℝ) → 𝕜) ‖z‖ : 𝕜))
h3 :
  HMul.hMul (α := 𝕜) (β := 𝕜) ((𝓚 : (a : ℝ) → 𝕜) ‖y‖ : 𝕜) ((𝓚 : (a : ℝ) → 𝕜) ‖y‖ : 𝕜) +
      HMul.hMul (α := 𝕜) (β := 𝕜) ((𝓚 : (a : ℝ) → 𝕜) ‖y + HSMul.hSMul (α := ℤ) 2 z‖ : 𝕜)
        ((𝓚 : (a : ℝ) → 𝕜) ‖y + HSMul.hSMul (α := ℤ) 2 z‖ : 𝕜) =
    2 *
      (HMul.hMul (α := 𝕜) (β := 𝕜) ((𝓚 : (a : ℝ) → 𝕜) ‖y + z‖ : 𝕜) ((𝓚 : (a : ℝ) → 𝕜) ‖y + z‖ : 𝕜) +
        HMul.hMul (α := 𝕜) (β := 𝕜) ((𝓚 : (a : ℝ) → 𝕜) ‖z‖ : 𝕜) ((𝓚 : (a : ℝ) → 𝕜) ‖z‖ : 𝕜))
h2 :
  HMul.hMul (α := 𝕜) (β := 𝕜) ((𝓚 : (a : ℝ) → 𝕜) ‖y + HSMul.hSMul (α := ℤ) (-2) z‖ : 𝕜)
        ((𝓚 : (a : ℝ) → 𝕜) ‖y + HSMul.hSMul (α := ℤ) (-2) z‖ : 𝕜) +
      HMul.hMul (α := 𝕜) (β := 𝕜) ((𝓚 : (a : ℝ) → 𝕜) ‖HSMul.hSMul (α := ℤ) 2 x + y‖ : 𝕜)
        ((𝓚 : (a : ℝ) → 𝕜) ‖HSMul.hSMul (α := ℤ) 2 x + y‖ : 𝕜) =
    2 *
      (HMul.hMul (α := 𝕜) (β := 𝕜) ((𝓚 : (a : ℝ) → 𝕜) ‖x + (y + HSMul.hSMul (α := ℤ) (-1) z)‖ : 𝕜)
          ((𝓚 : (a : ℝ) → 𝕜) ‖x + (y + HSMul.hSMul (α := ℤ) (-1) z)‖ : 𝕜) +
        HMul.hMul (α := 𝕜) (β := 𝕜) ((𝓚 : (a : ℝ) → 𝕜) ‖x + z‖ : 𝕜) ((𝓚 : (a : ℝ) → 𝕜) ‖x + z‖ : 𝕜))
h1 :
  HMul.hMul (α := 𝕜) (β := 𝕜) ((𝓚 : (a : ℝ) → 𝕜) ‖y + HSMul.hSMul (α := ℤ) 2 z‖ : 𝕜)
        ((𝓚 : (a : ℝ) → 𝕜) ‖y + HSMul.hSMul (α := ℤ) 2 z‖ : 𝕜) +
      HMul.hMul (α := 𝕜) (β := 𝕜) ((𝓚 : (a : ℝ) → 𝕜) ‖HSMul.hSMul (α := ℤ) 2 x + y‖ : 𝕜)
        ((𝓚 : (a : ℝ) → 𝕜) ‖HSMul.hSMul (α := ℤ) 2 x + y‖ : 𝕜) =
    2 *
      (HMul.hMul (α := 𝕜) (β := 𝕜) ((𝓚 : (a : ℝ) → 𝕜) ‖x + (y + z)‖ : 𝕜) ((𝓚 : (a : ℝ) → 𝕜) ‖x + (y + z)‖ : 𝕜) +
        HMul.hMul (α := 𝕜) (β := 𝕜) ((𝓚 : (a : ℝ) → 𝕜) ‖x + HSMul.hSMul (α := ℤ) (-1) z‖ : 𝕜)
          ((𝓚 : (a : ℝ) → 𝕜) ‖x + HSMul.hSMul (α := ℤ) (-1) z‖ : 𝕜))
⊢ 4⁻¹ *
      (HMul.hMul (α := 𝕜) (β := 𝕜) ((𝓚 : (a : ℝ) → 𝕜) ‖x + (y + z)‖ : 𝕜) ((𝓚 : (a : ℝ) → 𝕜) ‖x + (y + z)‖ : 𝕜) +
        (HSMul.hSMul (α := ℤ) (-1)
            (HMul.hMul (α := 𝕜) (β := 𝕜) ((𝓚 : (a : ℝ) → 𝕜) ‖x + (y + HSMul.hSMul (α := ℤ) (-1) z)‖ : 𝕜)
              ((𝓚 : (a : ℝ) → 𝕜) ‖x + (y + HSMul.hSMul (α := ℤ) (-1) z)‖ : 𝕜)) +
          (HMul.hMul (β := 𝕜)
              (HMul.hMul (α := 𝕜) (β := 𝕜) (I : 𝕜)
                ((𝓚 : (a : ℝ) → 𝕜)
                    ‖z + (HSMul.hSMul.{u_1, u_2, u_2} (α := 𝕜) I x + HSMul.hSMul.{u_1, u_2, u_2} (α := 𝕜) I y)‖ :
                  𝕜))
              ((𝓚 : (a : ℝ) → 𝕜)
                  ‖z + (HSMul.hSMul.{u_1, u_2, u_2} (α := 𝕜) I x + HSMul.hSMul.{u_1, u_2, u_2} (α := 𝕜) I y)‖ :
                𝕜) +
            HSMul.hSMul (α := ℤ) (-1)
              (HMul.hMul (β := 𝕜)
                (HMul.hMul (α := 𝕜) (β := 𝕜) (I : 𝕜)
                  ((𝓚 : (a : ℝ) → 𝕜)
                      ‖HSMul.hSMul (α := ℤ) (-1) z +
                          (HSMul.hSMul.{u_1, u_2, u_2} (α := 𝕜) I x + HSMul.hSMul.{u_1, u_2, u_2} (α := 𝕜) I y)‖ :
                    𝕜))
                ((𝓚 : (a : ℝ) → 𝕜)
                    ‖HSMul.hSMul (α := ℤ) (-1) z +
                        (HSMul.hSMul.{u_1, u_2, u_2} (α := 𝕜) I x + HSMul.hSMul.{u_1, u_2, u_2} (α := 𝕜) I y)‖ :
                  𝕜))))) =
    4⁻¹ *
        (HMul.hMul (α := 𝕜) (β := 𝕜) ((𝓚 : (a : ℝ) → 𝕜) ‖x + z‖ : 𝕜) ((𝓚 : (a : ℝ) → 𝕜) ‖x + z‖ : 𝕜) +
          (HSMul.hSMul (α := ℤ) (-1)
              (HMul.hMul (α := 𝕜) (β := 𝕜) ((𝓚 : (a : ℝ) → 𝕜) ‖x + HSMul.hSMul (α := ℤ) (-1) z‖ : 𝕜)
                ((𝓚 : (a : ℝ) → 𝕜) ‖x + HSMul.hSMul (α := ℤ) (-1) z‖ : 𝕜)) +
            (HMul.hMul (β := 𝕜)
                (HMul.hMul (α := 𝕜) (β := 𝕜) (I : 𝕜)
                  ((𝓚 : (a : ℝ) → 𝕜) ‖z + HSMul.hSMul.{u_1, u_2, u_2} (α := 𝕜) I x‖ : 𝕜))
                ((𝓚 : (a : ℝ) → 𝕜) ‖z + HSMul.hSMul.{u_1, u_2, u_2} (α := 𝕜) I x‖ : 𝕜) +
              HSMul.hSMul (α := ℤ) (-1)
                (HMul.hMul (β := 𝕜)
                  (HMul.hMul (α := 𝕜) (β := 𝕜) (I : 𝕜)
                    ((𝓚 : (a : ℝ) → 𝕜) ‖HSMul.hSMul (α := ℤ) (-1) z + HSMul.hSMul.{u_1, u_2, u_2} (α := 𝕜) I x‖ : 𝕜))
                  ((𝓚 : (a : ℝ) → 𝕜) ‖HSMul.hSMul (α := ℤ) (-1) z + HSMul.hSMul.{u_1, u_2, u_2} (α := 𝕜) I x‖ : 𝕜))))) +
      4⁻¹ *
        (HMul.hMul (α := 𝕜) (β := 𝕜) ((𝓚 : (a : ℝ) → 𝕜) ‖y + z‖ : 𝕜) ((𝓚 : (a : ℝ) → 𝕜) ‖y + z‖ : 𝕜) +
          (HSMul.hSMul (α := ℤ) (-1)
              (HMul.hMul (α := 𝕜) (β := 𝕜) ((𝓚 : (a : ℝ) → 𝕜) ‖y + HSMul.hSMul (α := ℤ) (-1) z‖ : 𝕜)
                ((𝓚 : (a : ℝ) → 𝕜) ‖y + HSMul.hSMul (α := ℤ) (-1) z‖ : 𝕜)) +
            (HMul.hMul (β := 𝕜)
                (HMul.hMul (α := 𝕜) (β := 𝕜) (I : 𝕜)
                  ((𝓚 : (a : ℝ) → 𝕜) ‖z + HSMul.hSMul.{u_1, u_2, u_2} (α := 𝕜) I y‖ : 𝕜))
                ((𝓚 : (a : ℝ) → 𝕜) ‖z + HSMul.hSMul.{u_1, u_2, u_2} (α := 𝕜) I y‖ : 𝕜) +
              HSMul.hSMul (α := ℤ) (-1)
                (HMul.hMul (β := 𝕜)
                  (HMul.hMul (α := 𝕜) (β := 𝕜) (I : 𝕜)
                    ((𝓚 : (a : ℝ) → 𝕜) ‖HSMul.hSMul (α := ℤ) (-1) z + HSMul.hSMul.{u_1, u_2, u_2} (α := 𝕜) I y‖ : 𝕜))
                  ((𝓚 : (a : ℝ) → 𝕜) ‖HSMul.hSMul (α := ℤ) (-1) z + HSMul.hSMul.{u_1, u_2, u_2} (α := 𝕜) I y‖ : 𝕜)))))

Tactic:
linear_combination (- h1 + h2 + h3 - h4 + I * (- h5 + h6 + h7 - h8)) / 8

Premises used: HMul.hMul, HSub.hSub, RCLike.I, HAdd.hAdd, Neg.neg