Assume that $x=\sum_{i=1}^nd_i\otimes d_i'$ is in the center of $D\otimes D'$, and assume that this presentation of $x$ as a sum of elementary tensors has a minimal number $n$ of terms. It follows that the elements $d_i', i=1,2,\ldots,n,$ are linearly independent over $F$, for otherwise we could reduce the number of terms in the obvious way. Let $r\in D$ be arbitrary. Using the fact that the sum of the subspaces $D\otimes d_i', i=1,2,\ldots,n$ is direct and comparing the products $(r\otimes 1)x$ and $x(r\otimes 1)$ shows that all the elements $d_i$ must commute with $r$. Therefore $d_i\in Z(D)=F$ for all $i=1,2,\ldots,n$. Therefore $n=1$. The rest is easy.