Let $K_1/K$ and $K_2/K$ two galois extension. I have a theorem that says that $K_1K_2/K$ is a galois extension and that if $K_1\cap K_2=K$ then $$\Gal(K_1K_2/K)\cong \Gal(K_1/K)\times \Gal(K_2/K).$$
I would like to generalize it for 3 extension field. i.e.
Let $K_1/K,K_2/K$ and $K_3/K$ three galois extension. Then, $K_1K_2K_3/K$ is galois and if $K_i\cap K_j=K$ for all $i\neq j$ then $$\Gal(K_1K_2K_3/K)\cong \Gal(K_1/K)\times \Gal(K_2/K)\times \Gal(K_3/K).$$
Now I tried to prove it using the first theorem below, but I can't conclude. The fact that $K_1K_2K_3/K$ is a galois extension is obvious. But for the rest, I really have difficulties.
My idea was to set $L=K_1K_2$, and thus show that $$\Gal(LK_3/K)\cong \Gal(L/K)\times \Gal(K_3/K),$$ using the 1st theorem below, but I can't prove the $L\cap K_3=K$.
Any idea ?