If $g$ is a left anti-derivative of the tuple $f$ ($leftDeriv a g = f a$ for every letter $a$) and each $f a$ is linearly finite, then $g$ is linearly finite. The witness is ${g} ∪ ⋃ₐ Gₐ$, where $Gₐ$ is a finite generator set for $f a$; the union is finite because $α$ is a $Fintype$. Every generator of the witness maps under a left derivative either to $f a$ (if it is $g$) or into the span of its own $Gₐ$ (if it comes from some $G_b$), and both of those lie in the span of the whole witness.