theorem member_of_left_coset_mk_left_coset_x_H_a_to_member_of_left_coset_mk_left_coset_y_H_b_to_member_of_left_coset_mk_left_coset_op_x_y_H_op_a_b:
∀G.∀H:normal_subgroup G.∀x,y,a,b.
a ∈ (x*H) → b ∈ (y*H) → (a·b) ∈ ((x·y)*H).
intros;
simplify;
theorem member_of_left_coset_mk_left_coset_x_H_a_to_member_of_left_coset_mk_left_coset_y_H_b_to_member_of_left_coset_mk_left_coset_op_x_y_H_op_a_b:
∀G.∀H:normal_subgroup G.∀x,y,a,b.
a ∈ (x*H) → b ∈ (y*H) → (a·b) ∈ ((x·y)*H).
intros;
simplify;