The Space of an Unsatisfiable Formula
Here is the original refutation:


From $a'+c$ and $a'+c'$ deduce $a'$.
From $a$ and $a'$ deduce $\square$.


When you add $b'$ back, you get:


From $a'+c$ and $a'+c'$ deduce $a'$.
From $a+b'$ and $a'$ deduce $b'$.


More generally, since the only difference between $\Gamma$ and $\Gamma'$ is the addition of $b'$, it is not hard to prove by induction that for every line $\ell$ proved in the refutation of $\Gamma$, the corresponding line in the corresponding $\Gamma'$ proof is either $\ell$ or $\ell+b'$. Hence the $\Gamma'$ proof proves either $\square$ or $b'$. It can happen that it indeed proves $\square$, in which case we can deduce $b'$ by weakening, which is an admissible rule for resolution space (we can get rid of it without increasing the space).