Since , the number of states in the above sequence must be greater than n + 1. But number of states in M is only n. hence, by pigeonhole principle at least one state must be repeated.

Let q_i and q_l be the q_l same state and is the first state to repeat in the sequence (there may be some more, that come later in the sequence). The sequence, now, looks like

which indicates that there must be sub strings x, y, z of w such that

This situation is depicted in the figure