-- Lean 4 proof step from Mathlib.LinearAlgebra.CliffordAlgebra.SpinGroup (goal)
Goal:
R : Type u_1
inst : CommRing R
M : Type u_2
inst_1 : AddCommGroup M
inst_2 : Module R M
Q : QuadraticForm R M
inst_3 : Invertible (α := R) 2
x : (CliffordAlgebra Q)ˣ
hx :
  Membership.mem (γ := Subgroup (CliffordAlgebra Q)ˣ)
    (Subgroup.closure (Units.val ⁻¹' Set.range (ι := M) (⇑(ι Q) : (a : M) → CliffordAlgebra Q))) x
b : M
mem :
  ∀ (x : (CliffordAlgebra Q)ˣ)
    (hx :
      Membership.mem (γ := Set (CliffordAlgebra Q)ˣ)
        (Units.val ⁻¹' Set.range (ι := M) (⇑(ι Q) : (a : M) → CliffordAlgebra Q)) x)
    (b : M),
    HMul.hMul (α := CliffordAlgebra Q) (β := CliffordAlgebra Q)
          ((involute : (a : CliffordAlgebra Q) → CliffordAlgebra Q) (↑x : CliffordAlgebra Q) : CliffordAlgebra Q)
          ((ι Q : (a : M) → CliffordAlgebra Q) b : CliffordAlgebra Q) *
        (↑x⁻¹ : CliffordAlgebra Q) ∈
      (ι Q).range
inv_mem :
  ∀ (x : (CliffordAlgebra Q)ˣ)
    (hx :
      Membership.mem (γ := Set (CliffordAlgebra Q)ˣ)
        (Units.val ⁻¹' Set.range (ι := M) (⇑(ι Q) : (a : M) → CliffordAlgebra Q)) x)
    (b : M),
    HMul.hMul (α := CliffordAlgebra Q) (β := CliffordAlgebra Q)
          ((involute : (a : CliffordAlgebra Q) → CliffordAlgebra Q) (↑x⁻¹ : CliffordAlgebra Q) : CliffordAlgebra Q)
          ((ι Q : (a : M) → CliffordAlgebra Q) b : CliffordAlgebra Q) *
        (↑x⁻¹⁻¹ : CliffordAlgebra Q) ∈
      (ι Q).range
one :
  ∀ (b : M),
    HMul.hMul (α := CliffordAlgebra Q) (β := CliffordAlgebra Q)
          ((involute : (a : CliffordAlgebra Q) → CliffordAlgebra Q) (↑1 : CliffordAlgebra Q) : CliffordAlgebra Q)
          ((ι Q : (a : M) → CliffordAlgebra Q) b : CliffordAlgebra Q) *
        (↑1⁻¹ : CliffordAlgebra Q) ∈
      (ι Q).range
⊢ HMul.hMul (α := CliffordAlgebra Q) (β := CliffordAlgebra Q)
        ((involute : (a : CliffordAlgebra Q) → CliffordAlgebra Q) (↑x : CliffordAlgebra Q) : CliffordAlgebra Q)
        ((ι Q : (a : M) → CliffordAlgebra Q) b : CliffordAlgebra Q) *
      (↑x⁻¹ : CliffordAlgebra Q) ∈
    (ι Q).range
have mul : ∀ (y z : (CliffordAlgebra Q)ˣ),
  Membership.mem (γ := Subgroup (CliffordAlgebra Q)ˣ)
      (Subgroup.closure (Units.val ⁻¹' Set.range (ι := M) (⇑(ι Q) : (a : M) → CliffordAlgebra Q))) y →
    Membership.mem (γ := Subgroup (CliffordAlgebra Q)ˣ)
        (Subgroup.closure (Units.val ⁻¹' Set.range (ι := M) (⇑(ι Q) : (a : M) → CliffordAlgebra Q))) z →
      ∀
        (hy :
          ∀ (b : M),
            HMul.hMul (α := CliffordAlgebra Q) (β := CliffordAlgebra Q)
                  ((involute : (a : CliffordAlgebra Q) → CliffordAlgebra Q) (↑y : CliffordAlgebra Q) :
                    CliffordAlgebra Q)
                  ((ι Q : (a : M) → CliffordAlgebra Q) b : CliffordAlgebra Q) *
                (↑y⁻¹ : CliffordAlgebra Q) ∈
              (ι Q).range)
        (hz :
          ∀ (b : M),
            HMul.hMul (α := CliffordAlgebra Q) (β := CliffordAlgebra Q)
                  ((involute : (a : CliffordAlgebra Q) → CliffordAlgebra Q) (↑z : CliffordAlgebra Q) :
                    CliffordAlgebra Q)
                  ((ι Q : (a : M) → CliffordAlgebra Q) b : CliffordAlgebra Q) *
                (↑z⁻¹ : CliffordAlgebra Q) ∈
              (ι Q).range)
        (b : M),
        HMul.hMul (α := CliffordAlgebra Q) (β := CliffordAlgebra Q)
              ((involute : (a : CliffordAlgebra Q) → CliffordAlgebra Q) (↑(y * z) : CliffordAlgebra Q) :
                CliffordAlgebra Q)
              ((ι Q : (a : M) → CliffordAlgebra Q) b : CliffordAlgebra Q) *
            (↑(y * z)⁻¹ : CliffordAlgebra Q) ∈
          (ι Q).range
Tactic: induction hx using Subgroup.closure_induction'' generalizing b with
  | mem x hx =>
    obtain ⟨a, ha⟩ := hx
    letI := x.invertible
    letI : Invertible (ι Q a) := by rwa [ha]
    letI : Invertible (Q a) := invertibleOfInvertibleι Q a
    simp_rw [← invOf_units x, ← ha, involute_ι, neg_mul, ι_mul_ι_mul_invOf_ι Q a b, ← map_neg,
      LinearMap.mem_range_self]
  | inv_mem x hx =>
    obtain ⟨a, ha⟩ := hx
    letI := x.invertible
    letI : Invertible (ι Q a) := by rwa [ha]
    letI : Invertible (Q a) := invertibleOfInvertibleι Q a
    letI := invertibleNeg (ι Q a)
    letI := Invertible.map involute (ι Q a)
    simp_rw [← invOf_units x, inv_inv, ← ha, map_invOf, involute_ι, invOf_neg, neg_mul,
      invOf_ι_mul_ι_mul_ι, ← map_neg, LinearMap.mem_range_self]
  | one => simp_rw [inv_one, Units.val_one, map_one, one_mul, mul_one, LinearMap.mem_range_self]
  | mul y z _ _ hy hz =>
    simp_rw [mul_inv_rev, Units.val_mul, map_mul]
    suffices involute (Q := Q) ↑y * (involute (Q := Q) ↑z * ι Q b * ↑z⁻¹) * ↑y⁻¹ ∈ _ by
      simpa only [mul_assoc] using this
    obtain ⟨z', hz'⟩ := hz b
    obtain ⟨y', hy'⟩ := hy z'
    simp_rw [← hz', ← hy', LinearMap.mem_range_self]
Cleared hypotheses: hx, b