I wasn't really sure what would be an appropriate title for this problem, but the statement is not too complicated.
I take $F/K$ a Galois extension of a field $K$. I take $\theta \in F$. Finally I take $N,N'$ two normal subextensions of $F$, such that $N'(\theta )$ is normal over $N'$. My question is whether the following inclusion is true:
$(N \cap N'(\theta )) \subseteqq (N \cap N')(\theta )$.