Let $\alpha$ be a finite sequence, and $|\alpha|$ be the length of $\alpha$.
Let $\alpha_1, \alpha_2$ be two finite sequences. Prove that $|\alpha_1 \alpha_2| = |\alpha_1 |+|\alpha_2|$. Where $\alpha_1 \alpha_2$ is just the concatenation of the sequences $\alpha_1$ and $\alpha_2$.
This looks very obvious. But I do not know what to do if I want to deal with it more precisely.