How do you prove this question about groups?

Question

Suppose that G is a group, H and K are subgroups of G such that H is a normal subgroup of G, HK=G and H ∩ K= {e}. Prove that each z ∈ G can be written uniquely, z=xy, where x ∈ H, y ∈ K.

Answer 1

Clearly for any $\;g\in G\;$ there exist $\;h_g\in H,\,k_g\in K\;$ s.t. $\;g=h_gk_g\;$ (why?) . If there were also $\;h\in H,\,k\in K\;$ with $\;g=hk\;$ , then:

$$hk=g=h_gk_g\implies h_g^{-1}h=k_gk^{-1}\in H\cap K$$

Write down now the final line of the proof...