-- Lean 4 proof step from Mathlib.Analysis.InnerProductSpace.OfNorm (goal)
Goal:
𝕜 : Type u_1
inst : RCLike 𝕜
E : Type u_2
inst_1 : NormedAddCommGroup E
inst_2 : NormedSpace 𝕜 E
x y : E
pos :
  ∀ (hI : Eq.{u_1 + 1} (α := 𝕜) I 0),
    4⁻¹ *
        (HMul.hMul (α := 𝕜) (↑‖x + y‖ : 𝕜) (↑‖x + y‖ : 𝕜) - HMul.hMul (α := 𝕜) (↑‖x - y‖ : 𝕜) (↑‖x - y‖ : 𝕜) +
            HMul.hMul (α := 𝕜) (-I) (↑‖HSMul.hSMul.{u_1, u_2, u_2} (α := 𝕜) I y + x‖ : 𝕜) *
              (↑‖HSMul.hSMul.{u_1, u_2, u_2} (α := 𝕜) I y + x‖ : 𝕜) -
          HMul.hMul (α := 𝕜) (-I) (↑‖HSMul.hSMul.{u_1, u_2, u_2} (α := 𝕜) I y - x‖ : 𝕜) *
            (↑‖HSMul.hSMul.{u_1, u_2, u_2} (α := 𝕜) I y - x‖ : 𝕜)) =
      4⁻¹ *
        (HMul.hMul (α := 𝕜) (β := 𝕜) ((𝓚 : (a : ℝ) → 𝕜) ‖x + y‖ : 𝕜) ((𝓚 : (a : ℝ) → 𝕜) ‖x + y‖ : 𝕜) -
              HMul.hMul (α := 𝕜) (β := 𝕜) ((𝓚 : (a : ℝ) → 𝕜) ‖x - y‖ : 𝕜) ((𝓚 : (a : ℝ) → 𝕜) ‖x - y‖ : 𝕜) +
            HMul.hMul (β := 𝕜)
              (HMul.hMul (α := 𝕜) (β := 𝕜) (I : 𝕜)
                ((𝓚 : (a : ℝ) → 𝕜) ‖HSMul.hSMul.{u_1, u_2, u_2} (α := 𝕜) I x + y‖ : 𝕜))
              ((𝓚 : (a : ℝ) → 𝕜) ‖HSMul.hSMul.{u_1, u_2, u_2} (α := 𝕜) I x + y‖ : 𝕜) -
          HMul.hMul (β := 𝕜)
            (HMul.hMul (α := 𝕜) (β := 𝕜) (I : 𝕜) ((𝓚 : (a : ℝ) → 𝕜) ‖HSMul.hSMul.{u_1, u_2, u_2} (α := 𝕜) I x - y‖ : 𝕜))
            ((𝓚 : (a : ℝ) → 𝕜) ‖HSMul.hSMul.{u_1, u_2, u_2} (α := 𝕜) I x - y‖ : 𝕜))
⊢ 4⁻¹ *
      (HMul.hMul (α := 𝕜) (↑‖x + y‖ : 𝕜) (↑‖x + y‖ : 𝕜) - HMul.hMul (α := 𝕜) (↑‖x - y‖ : 𝕜) (↑‖x - y‖ : 𝕜) +
          HMul.hMul (α := 𝕜) (-I) (↑‖HSMul.hSMul.{u_1, u_2, u_2} (α := 𝕜) I y + x‖ : 𝕜) *
            (↑‖HSMul.hSMul.{u_1, u_2, u_2} (α := 𝕜) I y + x‖ : 𝕜) -
        HMul.hMul (α := 𝕜) (-I) (↑‖HSMul.hSMul.{u_1, u_2, u_2} (α := 𝕜) I y - x‖ : 𝕜) *
          (↑‖HSMul.hSMul.{u_1, u_2, u_2} (α := 𝕜) I y - x‖ : 𝕜)) =
    4⁻¹ *
      (HMul.hMul (α := 𝕜) (β := 𝕜) ((𝓚 : (a : ℝ) → 𝕜) ‖x + y‖ : 𝕜) ((𝓚 : (a : ℝ) → 𝕜) ‖x + y‖ : 𝕜) -
            HMul.hMul (α := 𝕜) (β := 𝕜) ((𝓚 : (a : ℝ) → 𝕜) ‖x - y‖ : 𝕜) ((𝓚 : (a : ℝ) → 𝕜) ‖x - y‖ : 𝕜) +
          HMul.hMul (β := 𝕜)
            (HMul.hMul (α := 𝕜) (β := 𝕜) (I : 𝕜) ((𝓚 : (a : ℝ) → 𝕜) ‖HSMul.hSMul.{u_1, u_2, u_2} (α := 𝕜) I x + y‖ : 𝕜))
            ((𝓚 : (a : ℝ) → 𝕜) ‖HSMul.hSMul.{u_1, u_2, u_2} (α := 𝕜) I x + y‖ : 𝕜) -
        HMul.hMul (β := 𝕜)
          (HMul.hMul (α := 𝕜) (β := 𝕜) (I : 𝕜) ((𝓚 : (a : ℝ) → 𝕜) ‖HSMul.hSMul.{u_1, u_2, u_2} (α := 𝕜) I x - y‖ : 𝕜))
          ((𝓚 : (a : ℝ) → 𝕜) ‖HSMul.hSMul.{u_1, u_2, u_2} (α := 𝕜) I x - y‖ : 𝕜))
have neg : ∀ (hI : ¬Eq.{u_1 + 1} (α := 𝕜) I 0),
  4⁻¹ *
      (HMul.hMul (α := 𝕜) (↑‖x + y‖ : 𝕜) (↑‖x + y‖ : 𝕜) - HMul.hMul (α := 𝕜) (↑‖x - y‖ : 𝕜) (↑‖x - y‖ : 𝕜) +
          HMul.hMul (α := 𝕜) (-I) (↑‖HSMul.hSMul.{u_1, u_2, u_2} (α := 𝕜) I y + x‖ : 𝕜) *
            (↑‖HSMul.hSMul.{u_1, u_2, u_2} (α := 𝕜) I y + x‖ : 𝕜) -
        HMul.hMul (α := 𝕜) (-I) (↑‖HSMul.hSMul.{u_1, u_2, u_2} (α := 𝕜) I y - x‖ : 𝕜) *
          (↑‖HSMul.hSMul.{u_1, u_2, u_2} (α := 𝕜) I y - x‖ : 𝕜)) =
    4⁻¹ *
      (HMul.hMul (α := 𝕜) (β := 𝕜) ((𝓚 : (a : ℝ) → 𝕜) ‖x + y‖ : 𝕜) ((𝓚 : (a : ℝ) → 𝕜) ‖x + y‖ : 𝕜) -
            HMul.hMul (α := 𝕜) (β := 𝕜) ((𝓚 : (a : ℝ) → 𝕜) ‖x - y‖ : 𝕜) ((𝓚 : (a : ℝ) → 𝕜) ‖x - y‖ : 𝕜) +
          HMul.hMul (β := 𝕜)
            (HMul.hMul (α := 𝕜) (β := 𝕜) (I : 𝕜) ((𝓚 : (a : ℝ) → 𝕜) ‖HSMul.hSMul.{u_1, u_2, u_2} (α := 𝕜) I x + y‖ : 𝕜))
            ((𝓚 : (a : ℝ) → 𝕜) ‖HSMul.hSMul.{u_1, u_2, u_2} (α := 𝕜) I x + y‖ : 𝕜) -
        HMul.hMul (β := 𝕜)
          (HMul.hMul (α := 𝕜) (β := 𝕜) (I : 𝕜) ((𝓚 : (a : ℝ) → 𝕜) ‖HSMul.hSMul.{u_1, u_2, u_2} (α := 𝕜) I x - y‖ : 𝕜))
          ((𝓚 : (a : ℝ) → 𝕜) ‖HSMul.hSMul.{u_1, u_2, u_2} (α := 𝕜) I x - y‖ : 𝕜))
Tactic: by_cases hI : (I : 𝕜) = 0