Milne, on his notes about Class Field Theory (available in http://www.jmilne.org/math/CourseNotes/CFT.pdf, on page 47 more precisely), given a local field $K$, constructs its maximal unramified extension contained in its separable closure, $K^{un}$, and, give a prime element $\pi$ of $K$, constructs the extension $K_\pi$, which is the union of totally ramified extensions $K_{\pi,n}$ of degree $n$, using Lubin-Tate theory. Then he argues that $K^{ab}=K^{un}\cdot K_\pi$ (local kronecker-weber), where $K^{ab}$ is the maximal abelian extension.
One of the lemmas he uses to prove this result is the following:
A finite unramified extension $L$ of $K_\pi$ is contained in $K_\pi\cdot K^{un}$.
Here is its proof: $L$ must be of the form $K_\pi\cdot L^{\prime}$, where $L^{\prime}$ is some unramified extension of $K_{\pi,n}$ for some $n$. Also, $L^{\prime}$ must be of the form $K_{\pi,n}\cdot L^{\prime \prime}$, where $L^{\prime \prime} $ is some unramified extension of $K$.
So, my question is: how do you prove the following assertions:
(1) $L$ must be of the form $K_\pi\cdot L^{\prime}$;
(2) $L^{\prime}$ must be of the form $K_{\pi,n}\cdot L^{\prime \prime}$?
Thank you.
This question is answered by the comments above, so the topic can be closed.