have hsubset : (s ∩ a • C.mulStab * (t ∩ b • C.mulStab)).mulStab ⊆ C.mulStab := by
  have hxymem : x * y ∈ s ∩ a • C.mulStab * (t ∩ b • C.mulStab) := mul_mem_mul hx hy
  apply subset_trans (mulStab_subset_div_right hxymem)
  have : s ∩ a • C.mulStab * (t ∩ b • C.mulStab) ⊆ (x * y) • C.mulStab := by
    apply subset_trans (mul_subset_mul inter_subset_right inter_subset_right)
    rw [smul_mul_smul]
    rw [← hac, ← had, smul_mul_smul, smul_assoc]
    apply smul_finset_subset_smul_finset
    rw [← smul_smul]
    rw [mul_subset_iff]
    intro x hx y hy
    rw [smul_mulStab hd, smul_mulStab hc, mem_mulStab hCne, ← smul_smul,
      (mem_mulStab hCne).mp hy, (mem_mulStab hCne).mp hx]
  apply subset_trans (div_subset_div_right this) _
  have hsing : (x * y) • C.mulStab = {x * y} * C.mulStab := by rw [singleton_mul]; rfl
  simp_rw [hsing, singleton_mul, div_singleton, image_image, div_eq_mul_inv, mul_comm, comp,
    mul_inv_cancel_right, image_id', subset_refl]
case intro.intro.intro.intro.intro.intro
α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s s' t t' C : Finset α
a b : α
hab : ¬(a * b) • C.mulStab ⊆ s * t
hCne : C.Nonempty
x : α
hx : x ∈ s ∩ a • C.mulStab
y : α
hy : y ∈ t ∩ b • C.mulStab
c : α
hc : c ∈ C.mulStab
hac : a • c = x
d : α
hd : d ∈ C.mulStab
had : b • d = y
⊢ (s ∩ a • C.mulStab * (t ∩ b • C.mulStab)).mulStab ⊂ C.mulStab
case intro.intro.intro.intro.intro.intro
α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s s' t t' C : Finset α
a b : α
hab : ¬(a * b) • C.mulStab ⊆ s * t
hCne : C.Nonempty
x : α
hx : x ∈ s ∩ a • C.mulStab
y : α
hy : y ∈ t ∩ b • C.mulStab
c : α
hc : c ∈ C.mulStab
hac : a • c = x
d : α
hd : d ∈ C.mulStab
had : b • d = y
hsubset : (s ∩ a • C.mulStab * (t ∩ b • C.mulStab)).mulStab ⊆ C.mulStab
⊢ (s ∩ a • C.mulStab * (t ∩ b • C.mulStab)).mulStab ⊂ C.mulStab