For a language $L$ with pumping length $p$, and a string $s\in L$, the pumping lemmas are as follows:
Regular version: If $|s| \geq p$, then $s$ can be written as $xyz$, satisfying the following conditions:
- $|y|\geq 1$
- $|xy|\leq p$
- $ \forall i\geq 0: xy^iz\in L$
Context-free version: If $|s| \geq p$, then $s$ can be written as $uvxyz$, satisfying the following conditions:
- $|vy|\geq 1$
- $|vxy|\leq p$
- $ \forall i\geq 0: uv^ixy^iz\in L$
My question is this: Why do we have condition 2 in the lemma (for either case)? I understand that condition 1 essentially says that the "pumpable" (meaning nullable or arbitrarily repeatable) substring has to have some nonzero length, and condition 3 says that the pumpable substring can be repeated arbitrarily many times without deriving an invalid string (with respect to $L$). I'm not sure what the second condition means or why it is important. Is there a simple but meaningful example to illustrate its importance?