Showing that from two cut-free derivations $\Gamma \vdash A$ and $\Gamma, A \vdash C$ you can produce a cut-free derivation of $\Gamma \vdash C$ is called cut admissibility.
Admissibility of cut is actually equivalent to showing that cut can be eliminated. If you have cut-elimination, you just cut the two derivations together and then eliminate the cut. On the other hand, if you have cut-admissibility, you can use it to recursively eliminate all the cuts, starting at the leaves and working down to the root.
A good description of this proof, for a variety of logics, can be found in Frank Pfenning's 1995 LICS paper, Structural Cut Elimination.