Derivation of cut rule in sequent calculus
The fact that a system is complete for proving valid formulas without the cut rule doesn't mean you can derive the cut rule from other rules. 

In fact it is not difficult to construct counter-examples. Consider 
$\Rightarrow A \rightarrow B$ and $\Rightarrow A$. From these assumptions it would follow that $\Rightarrow B$. But you cannot derive it without using a cut.