/-- If a group has `UniqueProds`, then it actually has `TwoUniqueProds`.
  For an example of a semigroup `G` embeddable into a group that has `UniqueProds`
  but not `TwoUniqueProds`, see Example 10.13 in
  [J. Okniński, *Semigroup Algebras*][Okninski1991]. -/
@[to_additive] theorem toTwoUniqueProds_of_group {G}
    [Group G] [UniqueProds G] : TwoUniqueProds G where
  uniqueMul_of_one_lt_card {A B} hc := by
    simp_rw [Nat.one_lt_mul_iff, card_pos] at hc
    obtain ⟨a, ha, b, hb, hu⟩ := uniqueMul_of_nonempty hc.1 hc.2.1
    let C := A.map ⟨_, mul_right_injective a⁻¹⟩ -- C = a⁻¹A
    let D := B.map ⟨_, mul_left_injective b⁻¹⟩  -- D = Bb⁻¹
    have hcard : 1 < C.card ∨ 1 < D.card; · simp_rw [card_map]; exact hc.2.2
    have hC : 1 ∈ C := mem_map.mpr ⟨a, ha, inv_mul_self a⟩
    have hD : 1 ∈ D := mem_map.mpr ⟨b, hb, mul_inv_self b⟩
    suffices : ∃ c ∈ C, ∃ d ∈ D, (c ≠ 1 ∨ d ≠ 1) ∧ UniqueMul C D c d
    · simp_rw [mem_product]
      obtain ⟨c, hc, d, hd, hne, hu'⟩ := this
      obtain ⟨a0, ha0, rfl⟩ := mem_map.mp hc
      obtain ⟨b0, hb0, rfl⟩ := mem_map.mp hd
      refine ⟨(_, _), ⟨ha0, hb0⟩, (a, b), ⟨ha, hb⟩, ?_, fun a' b' ha' hb' he => ?_, hu⟩
      simp_rw [Function.Embedding.coeFn_mk, Ne, inv_mul_eq_one, mul_inv_eq_one] at hne
      · rwa [Ne, Prod.mk.inj_iff, not_and_or, eq_comm]
      specialize hu' (mem_map_of_mem _ ha') (mem_map_of_mem _ hb')
      simp_rw [Function.Embedding.coeFn_mk, mul_left_cancel_iff, mul_right_cancel_iff] at hu'
      rw [mul_assoc, ← mul_assoc a', he, mul_assoc, mul_assoc] at hu'
      exact hu' rfl
    classical
    let _ := Finset.mul (α := G)              -- E = D⁻¹C, F = DC⁻¹
    have := uniqueMul_of_nonempty (A := D.image (·⁻¹) * C) (B := D * C.image (·⁻¹)) ?_ ?_
    · obtain ⟨e, he, f, hf, hu⟩ := this
      clear_value C D
      simp only [UniqueMul, mem_mul, mem_image] at he hf hu
      obtain ⟨_, c1, ⟨d1, hd1, rfl⟩, hc1, rfl⟩ := he
      obtain ⟨d2, _, hd2, ⟨c2, hc2, rfl⟩, rfl⟩ := hf
      by_cases h12 : c1 ≠ 1 ∨ d2 ≠ 1
      · refine ⟨c1, hc1, d2, hd2, h12, fun c3 d3 hc3 hd3 he => ?_⟩
        specialize hu ⟨_, _, ⟨_, hd1, rfl⟩, hc3, rfl⟩ ⟨_, _, hd3, ⟨_, hc2, rfl⟩, rfl⟩
        rw [mul_left_cancel_iff, mul_right_cancel_iff,
            mul_assoc, ← mul_assoc c3, he, mul_assoc, mul_assoc] at hu; exact hu rfl
      push_neg at h12; obtain ⟨rfl, rfl⟩ := h12
      by_cases h21 : c2 ≠ 1 ∨ d1 ≠ 1
      · refine ⟨c2, hc2, d1, hd1, h21, fun c4 d4 hc4 hd4 he => ?_⟩
        specialize hu ⟨_, _, ⟨_, hd4, rfl⟩, hC, rfl⟩ ⟨_, _, hD, ⟨_, hc4, rfl⟩, rfl⟩
        simpa only [mul_one, one_mul, ← mul_inv_rev, he, true_imp_iff, inv_inj, and_comm] using hu
      push_neg at h21; obtain ⟨rfl, rfl⟩ := h21
      rcases hcard with hC | hD
      · obtain ⟨c, hc, hc1⟩ := exists_ne_of_one_lt_card hC 1
        refine (hc1 ?_).elim
        simpa using hu ⟨_, _, ⟨_, hD, rfl⟩, hc, rfl⟩ ⟨_, _, hD, ⟨_, hc, rfl⟩, rfl⟩
      · obtain ⟨d, hd, hd1⟩ := exists_ne_of_one_lt_card hD 1
        refine (hd1 ?_).elim
        simpa using hu ⟨_, _, ⟨_, hd, rfl⟩, hC, rfl⟩ ⟨_, _, hd, ⟨_, hC, rfl⟩, rfl⟩
    

Proof state:
case this.refine_1
G✝ : Type u
H : Type v
inst✝³ : Mul G✝
inst✝² : Mul H
G : Type u_1
inst✝¹ : Group G
inst✝ : UniqueProds G
A B : Finset G
hc : Finset.Nonempty A ∧ Finset.Nonempty B ∧ (1 < card A ∨ 1 < card B)
a : G
ha : a ∈ A
b : G
hb : b ∈ B
hu : UniqueMul A B a b
C : Finset G := map { toFun := fun x => a⁻¹ * x, inj' := (_ : Function.Injective fun x => a⁻¹ * x) } A
D : Finset G := map { toFun := fun x => x * b⁻¹, inj' := (_ : Function.Injective fun x => x * b⁻¹) } B
hcard : 1 < card C ∨ 1 < card D
hC : 1 ∈ C
hD : 1 ∈ D
x✝ : Mul (Finset G) := Finset.mul
⊢ Finset.Nonempty (image (fun x => x⁻¹) D * C)
case this.refine_2
G✝ : Type u
H : Type v
inst✝³ : Mul G✝
inst✝² : Mul H
G : Type u_1
inst✝¹ : Group G
inst✝ : UniqueProds G
A B : Finset G
hc : Finset.Nonempty A ∧ Finset.Nonempty B ∧ (1 < card A ∨ 1 < card B)
a : G
ha : a ∈ A
b : G
hb : b ∈ B
hu : UniqueMul A B a b
C : Finset G := map { toFun := fun x => a⁻¹ * x, inj' := (_ : Function.Injective fun x => a⁻¹ * x) } A
D : Finset G := map { toFun := fun x => x * b⁻¹, inj' := (_ : Function.Injective fun x => x * b⁻¹) } B
hcard : 1 < card C ∨ 1 < card D
hC : 1 ∈ C
hD : 1 ∈ D
x✝ : Mul (Finset G) := Finset.mul
⊢ Finset.Nonempty (D * image (fun x => x⁻¹) C)

Next tactic:
all_goals apply_rules [Nonempty.mul, Nonempty.image, Finset.Nonempty.map, hc.1, hc.2.1]