Learning class field theory I found this theorem, but I can't prove it or find the solution. I'll be glad to any help.
Let $L$ and $M$ be abelian extensions of $K$. $L \subset M$ if and only if there is a modulus $\mathfrak m$, divisible by all primes of $K$ ramified in either $L$ or $M$, such that $$ P_{K,1}(\mathfrak m)\subset \ker(\Phi_{M/K,\mathfrak m}) \subset \ker(\Phi_{L/K,\mathfrak m}).$$
$\Phi_{M/K,\mathfrak m}$ - is Artin map for modulus $\mathfrak m$. $P_{K,1}$ is a subgroup of the group of fractions ideals, generated by principal $\alpha \mathcal O_K$-ideals, where $\alpha$ satisfies $\alpha \equiv 1 \pmod{\mathfrak m_0}$ and $\sigma (\alpha) > 0$ for every real infinite prime $\sigma$ dividing infinite part of $\mathfrak m.$
It's quite easy to prove that $L \subset M$ implies $$ P_{K,1}(\mathfrak m)\subset \ker(\Phi_{M/K,\mathfrak m}) \subset \ker(\Phi_{L/K,\mathfrak m}).$$ But I don't know how to prove another implication.