-- Lean 4 proof step from Mathlib.Algebra.Group.UniqueProds.Basic (goal)
Goal:
G_1 : Type u
H : Type v
inst : Mul G_1
inst_1 : Mul H
ι : Type u_2
G : ι → Type u_1
inst_2 : (i : ι) → Mul (G i)
inst_3 : ∀ (i : ι), TwoUniqueProds.{u_1} (G i)
A : Finset ((i : ι) → G i)
let let₁ let₂ :
  ∀ {B : Finset ((i : ι) → G i)},
    1 < #A * #B → ∃ p1 ∈ A ×ˢ B, ∃ p2 ∈ A ×ˢ B, p1 ≠ p2 ∧ UniqueMul A B p1.1 p1.2 ∧ UniqueMul A B p2.1 p2.2
let₃ let₄ :
  ∀ (x : Finset ((i : ι) → G i)),
    (∀ y ⊂ x,
        ∀ {B : Finset ((i : ι) → G i)},
          1 < #y * #B → ∃ p1 ∈ y ×ˢ B, ∃ p2 ∈ y ×ˢ B, p1 ≠ p2 ∧ UniqueMul y B p1.1 p1.2 ∧ UniqueMul y B p2.1 p2.2) →
      ∀ {B : Finset ((i : ι) → G i)},
        1 < #x * #B → ∃ p1 ∈ x ×ˢ B, ∃ p2 ∈ x ×ˢ B, p1 ≠ p2 ∧ UniqueMul x B p1.1 p1.2 ∧ UniqueMul x B p2.1 p2.2
let₅ :
  ∀ (A : Finset ((i : ι) → G i))
    (ihA :
      ∀ y ⊂ A,
        ∀ {B : Finset ((i : ι) → G i)},
          1 < #y * #B → ∃ p1 ∈ y ×ˢ B, ∃ p2 ∈ y ×ˢ B, p1 ≠ p2 ∧ UniqueMul y B p1.1 p1.2 ∧ UniqueMul y B p2.1 p2.2)
    {B : Finset ((i : ι) → G i)},
    1 < #A * #B → ∃ p1 ∈ A ×ˢ B, ∃ p2 ∈ A ×ˢ B, p1 ≠ p2 ∧ UniqueMul A B p1.1 p1.2 ∧ UniqueMul A B p2.1 p2.2
let₆ let₇ let₈ :
  ∀ (A : Finset ((i : ι) → G i))
    (ihA :
      ∀ y ⊂ A,
        ∀ {B : Finset ((i : ι) → G i)},
          1 < #y * #B → ∃ p1 ∈ y ×ˢ B, ∃ p2 ∈ y ×ˢ B, p1 ≠ p2 ∧ UniqueMul y B p1.1 p1.2 ∧ UniqueMul y B p2.1 p2.2)
    {B : Finset ((i : ι) → G i)} (B : Finset ((i : ι) → G i))
    (ihB :
      ∀ y ⊂ B, 1 < #A * #y → ∃ p1 ∈ A ×ˢ y, ∃ p2 ∈ A ×ˢ y, p1 ≠ p2 ∧ UniqueMul A y p1.1 p1.2 ∧ UniqueMul A y p2.1 p2.2)
    (hc : 1 < #A * #B), ∃ p1 ∈ A ×ˢ B, ∃ p2 ∈ A ×ˢ B, p1 ≠ p2 ∧ UniqueMul A B p1.1 p1.2 ∧ UniqueMul A B p2.1 p2.2
⊢ ∀ {B : Finset ((i : ι) → G i)},
    1 < #A * #B → ∃ p1 ∈ A ×ˢ B, ∃ p2 ∈ A ×ˢ B, p1 ≠ p2 ∧ UniqueMul A B p1.1 p1.2 ∧ UniqueMul A B p2.1 p2.2
have let₉ : ∀ (A : Finset ((i : ι) → G i))
  (ihA :
    ∀ y ⊂ A,
      ∀ {B : Finset ((i : ι) → G i)},
        1 < #y * #B → ∃ p1 ∈ y ×ˢ B, ∃ p2 ∈ y ×ˢ B, p1 ≠ p2 ∧ UniqueMul y B p1.1 p1.2 ∧ UniqueMul y B p2.1 p2.2)
  {B : Finset ((i : ι) → G i)} (B : Finset ((i : ι) → G i))
  (ihB :
    ∀ y ⊂ B, 1 < #A * #y → ∃ p1 ∈ A ×ˢ y, ∃ p2 ∈ A ×ˢ y, p1 ≠ p2 ∧ UniqueMul A y p1.1 p1.2 ∧ UniqueMul A y p2.1 p2.2)
  (hc : 1 < #A * #B) (hA : 0 < #A) (hB : 0 < #B) (hc : 1 < #A ∨ 1 < #B),
  ∃ p1 ∈ A ×ˢ B, ∃ p2 ∈ A ×ˢ B, p1 ≠ p2 ∧ UniqueMul A B p1.1 p1.2 ∧ UniqueMul A B p2.1 p2.2
Tactic: classical
    let _ := isWellFounded_ssubset (α := ∀ i, G i) -- why need this?
    apply IsWellFounded.induction (· ⊂ ·) A; intro A ihA B
    apply IsWellFounded.induction (· ⊂ ·) B; intro B ihB hc
    obtain ⟨hA, hB, hc⟩ := Nat.one_lt_mul_iff.mp hc
    rw [card_pos] at hA hB
    obtain ⟨i, hc⟩ := exists_or.mpr (hc.imp exists_of_one_lt_card_pi exists_of_one_lt_card_pi)
    obtain ⟨p1, h1, p2, h2, hne, hi1, hi2⟩ := uniqueMul_of_one_lt_card (Nat.one_lt_mul_iff.mpr
      ⟨card_pos.2 (hA.image _), card_pos.2 (hB.image _), hc.imp And.left And.left⟩)
    simp_rw [mem_product, mem_image, ← filter_nonempty_iff] at h1 h2
    replace h1 := uniqueMul_of_twoUniqueMul ?_ h1.1 h1.2
    on_goal 1 => replace h2 := uniqueMul_of_twoUniqueMul ?_ h2.1 h2.2
    · obtain ⟨a1, ha1, b1, hb1, hu1⟩ := h1
      obtain ⟨a2, ha2, b2, hb2, hu2⟩ := h2
      rw [mem_filter] at ha1 hb1 ha2 hb2
      simp_rw [mem_product]
      refine ⟨(a1, b1), ⟨ha1.1, hb1.1⟩, (a2, b2), ⟨ha2.1, hb2.1⟩, ?_,
        UniqueMul.of_image_filter (Pi.evalMulHom G i) ha1.2 hb1.2 hi1 hu1,
        UniqueMul.of_image_filter (Pi.evalMulHom G i) ha2.2 hb2.2 hi2 hu2⟩
      grind
    all_goals rcases hc with hc | hc; · exact ihA _ (hc.2 _)
    · by_cases hA : {a ∈ A | a i = p2.1} = A
      · rw [hA]
        exact ihB _ (hc.2 _)
      · exact ihA _ ((A.filter_subset _).ssubset_of_ne hA)
    · by_cases hA : {a ∈ A | a i = p1.1} = A
      · rw [hA]
        exact ihB _ (hc.2 _)
      · exact ihA _ ((A.filter_subset _).ssubset_of_ne hA)