apply pow_gcd_eq_one
case refine'_2
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 ^ gcd (Monoid.exponent ↥A) (Monoid.exponent ↥B) = 1
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

case refine'_2.hn
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 ↥B = 1