I am using the following definition of a normal extension: $K \subset L$ is normal if for all $\Omega$ with $K \subset \Omega$, for all $K$ embeddings $x_1:L\rightarrow \Omega, x_2:L\rightarrow \Omega$, we have that $x_1(L) \subset x_2(L)$.
I was given a proof that splitting fields are normal which goes as follows: $K \subset L$ a splitting field for some $f(x) \in K[X]$. Then, it is immediately assumed that $L \subset \Omega$, $x:L \rightarrow \Omega$ is a $K$-embedding, and then shown that $x(L)=L$. This concludes the proof (I didn't add much detail to this hoping it's not necessary for my question). I understand most of what is going on in the proof, the only issue is this assumption that we can take $L \subset \Omega$. The definition states that the property must hold for all $\Omega$ such that $K \subset \Omega$, however we show it for all $\Omega$ such that $L \subset \Omega$. How is it possible to make such a reduction?