Covariance formula with CDF (Hoeffding's Covariance Identity)

A complete proof of above lemma can be found on page 241 (Lemma 7.27) of Quantitative Risk Management: Concepts, Techniques and Tools.

Hint: 2$cov(X_1, X_2) = E[(X_1-\tilde{X_1})(X_2-\tilde{X_2})]$,
where $(\tilde{X_1}, \tilde{X_2})$ is an independent copy with the same joint distribution function as  $(X_1, X_2)$.

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