# Explicit form of Klein factors in Giamarchi

In Giamarchi, Quantum Physics in One Dimension, Appendix B, I don't understand how he did his last step in equation B.8, as shown below.

If anyone has gone over the derivation, I would really appreciate if you could elaborate on the final equality.

EDIT: I realized that the Kronecker delta is nonzero only if $$k$$ is one of the filled states for nonzero $$n_p$$. In that case, the operator $$a_{rk}^*$$ contributes zero to the summation and thus only the zeros order, i.e., $$n_p=0$$ for all $$rp>0$$, has a nonzero contribution to the summation.