How do we show that $\displaystyle A \vdash_L \neg \neg A$withoutgiving a derivation?

I know that the derivation for $\displaystyle \neg \neg A \vdash_L A$ is trivial, but the derivation of $\displaystyle A \vdash_L \neg \neg A$ is pretty long. But the question is how to show thatwithoutpresenting a derivation. How can we do that? Any help and suggestions are greatly appreciated.

The system L: http://img405.imageshack.us/img405/4208/systemli.jpg