In paragraph V.1 of Algebra proposition 1.7 Lang claims that the class of algebraic extensions is distinguished. I know that if $F/k$ and $E/F$ are algebraic extensions than so is the $E/k$ - that is easy to prove. However, the second required property is that if $E/k$ is algebraic and $F/k$ is arbitrary, then (assuming the compositum is defined) $EF/F$ is algebraic. Lang more or less skips the proof of this, only saying that "an element remains algebraic under lifting, and hence does the extension."
However, there are no finiteness conditions here and no element given in advance. Although $EF$ is generated over $F$ by a set of elements which are algebraic over $F$ (namely, all the elements of $E$), there are no facts known (up to this point in the book at least) about the infinitely generated extensions.
What am I missing?