Let $K_1$ and $K_2$ be subfields of an algebraically closed field $\Omega$ and $E$ a common subfield of both $K_1$ and $K_2$. We say that $K_1$ and $K_2$ are linearly disjoint over $E$ if every tuple $\bar{a}$ in $K_1$ that is linearly independent over $E$, remains linearly independent over $K_2$. Similarly, we say that $K_1$ and $K_2$ are free over $E$ if every tuple $\bar{a}$ in $K_1$ that is algebraically independent over $E$ remains algebraically independent over $E$.
In some lecture notes I am studying, they state that if $E$ is an algebraically closed field, then the two notions above agree, but no proof or reference is given. Why is this true?