Can I exchange the indices in this particular product of Kronecker delta's?

Question

Is the following true? (We are not using Einstein summation notation here)

$$\delta_{mk} \delta_{nk} = \delta_{mn} \delta_{km}$$

Answer 1

Yes, this is true. Not, however, as your title might seem to suggest, because you can generally exchange indices on pairs of Kronecker symbols, but because in this specific case the products on both sides enforce $m=k=n$, and it doesn’t matter whether you do that in the form $m=k$, $n=k$ or $m=n$, $k=m$.