Since 
is the first repeated state, we have,
and at the same time
y cannot be empty i.e
.
From the above, it immediately follows that
. Hence 
.
Similarly,
implying 
implying 
and so on.
That is, starting at the loop on state can be omitted, taken once, twice, or many more times, (by the DFA M ) eventually arriving at the final state
Thus, accepting the string xz, xyz, xy2z,... i.e. xyiz for all 
Hence 
.