rw [← Subgroup.coe_mk A x hx.1, ← Subgroup.coe_pow, Monoid.pow_exponent_eq_one,
  Subgroup.coe_one]
case refine'_2.hm
G : Type u_1
inst✝¹ : Group G
inst✝ : Fintype G
A B : Subgroup G
h : gcd (Monoid.exponent ↥A) (Monoid.exponent ↥B) = 1
x : G
hx : x ∈ A ∧ x ∈ B
⊢ x ^ Monoid.exponent ↥A = 1
no goals