intro i hi
case a
G : Type u_1
inst✝ : Group G
X : Set G
x : G
hx : ∃ x_1, (∀ x ∈ x_1, x ∈ X ∪ X⁻¹) ∧ List.prod x_1 = x
⊢ ∀ i ∈ {K | X ⊆ ↑K}, x ∈ i
case a
G : Type u_1
inst✝ : Group G
X : Set G
x : G
hx : ∃ x_1, (∀ x ∈ x_1, x ∈ X ∪ X⁻¹) ∧ List.prod x_1 = x
i : Subgroup G
hi : i ∈ {K | X ⊆ ↑K}
⊢ x ∈ i