have hmul_mem : ∀ a b : α, a ∈ whole_group → b ∈ whole_group → a * b ∈ whole_group := by
  rw [whole_group]
  intros a b _ _
  rw [Membership.mem, instMembershipSet]
  simp
α : Type u_1
G : group α
⊢ subgroup G whole_group
α : Type u_1
G : group α
hmul_mem : ∀ (a b : α), a ∈ whole_group → b ∈ whole_group → a * b ∈ whole_group
⊢ subgroup G whole_group