In this paper, there is a proof of Hoeffding identity for covariance (see page 541 or page 5 in pdf file, Theorem 1.11). A part of the proof is the following equality:
$ 2\text{cov}(X,Y) = 2(E(XY)-E(X)E(Y))=E((X_1-X_2)(Y_1 - Y_2))$
As far as I understand this transition is done by taking into account that$\ (X_1,Y_1)$ is iid copy of$\ (X_2,Y_2)$. It should be pretty simple, but I'm having hard time writing this explicitly. Can you please give me a hint?