show that if L is a regular, then drop(L) is a regular
I'm not sure what $L1, \dots, Lk$ are since you did not define them. The easiest way is probably that of starting with a DFA for $L$ and constructing a NFA for $drop(L)$ (hint in the spoiler below).


   Replace all transitions labelled with "a" in the DFA for $L$ with $\epsilon$-transitions in the NFA for $drop(L)$.


Then it should be easy to show that:


If $w \in drop(L)$ then the NFA accepts $w$: use the definition of the function $drop$ to conclude that there must be a suitable word $w' \in L$.  Start from an accepting path $\pi$ for $w'$ in the DFA for $L$ and modify $\pi$ to be an accepting path for $w$ in the NFA for $drop(L)$.
If the NFA accepts a word $w$ then $w \in drop(L)$: look at an accepting path in the NFA for $drop(L)$ and use it construct a word $w'$ that is accepted by the DFA for $L$. Use the fact that you just proved that $w' \in L$ and the definition of the function $drop$ to conclude that $w$ must belong to $drop(L)$.


I'm purposefully being a bit vague on the details, so I don't completely give away the solution. Comment if you need more help.