I'm going through 'Proof Theory' by Gaisi Takeuti (second edition 1987). Theorem 6.9 on page 34 talks about when one can replace all occurrences of a term $t$ in a provable sequent $S$ by a free variable to get another provable sequent $S'$. The sufficient condition stated in the theorem is that there's no sub-semi-term of $t$ is $S$.
My question is: what examples exist where the existence of a sub-semi-term of $t$ in $S$ prevents a proof of $S'$? As far as I can tell, one can still construct a proof $P'$ of $S'$ from the original proof $P$, albeit with possibly partial replacement of $t$ in sequents other than the lowest one.