Prove the following generalization of the pumping lemma, which can sometimes make it unnecessary to break the proof into cases. If L can be accepted by an FA, then there is an integer n such that for any x ∈ L, and any way of writing x as x = x₁x₂x₃ with |x₂|=n, there are strings u, v, and w such that

a. x2 = uvw

b. |v| > 0

c. For every m ≥ 0, x1uvmwx3 ∈ L.