Exercise 2.5 in Vijay Krishna's Auction Theory (slightly modified):
Consider a two-bidder ﬁrst-price auction in which bidders’ values are distributed according to $F$. Let $\beta$ be the symmetric equilibrium, i.e., $$\beta(x)=E[Y\mid Y<x],$$ where $Y$ is drawn from $F$.
Now suppose that after the auction is over, both the losing and winning bids are publicly announced. In addition, there is the possibility of postauction resale: The winner of the auction may, if he so wishes, offer the object to the other bidder at a ﬁxed “take-it-or-leave-it” price of $p$. If the other bidder agrees, then the object changes hands, and the losing bidder pays the winning bidder $p$. Otherwise, the object stays with the winning bidder, and no money changes hands. The possibility of postauction resale in this manner is commonly known to both bidders prior to participating in the auction. Show that $\beta$ remains an equilibrium even if resale is allowed. In particular, show that a bidder with value $x$ cannot gain by bidding an amount $b>\beta(x)$ even when he has the option of reselling the object to the other bidder.
Call the two bidders Jane and Mike. Given Mike's bid, and assuming that Mike follows the equilibrium strategy $\beta$, Jane can determine Mike's exact value. This implies that if Jane wins the item but realizes that she has lower value than Mike, she will resell it to Mike at exactly Mike's value.
We should calculate Jane's expected payoff when bidding $\beta(z)$ with $z\geq x$ when her value is $x$, and show that the expected payoff is maximized when $z=x$.
Jane wins with probability $F(z)$. If she loses, her payoff is $0$. If she wins and her value is higher than Mike's value, her payoff is $x-\beta(z)$. If she wins and her value is lower than Mike's value, $Y$, her payoff is $y-\beta(z)$. As a result, her expected payoff is $$-F(z)\beta(z)+F(x)x+(F(z)-F(x))E[Y\mid x<Y<z].$$
The term $F(x)x$ does not depend on $z$, so we are left to maximize $$-F(z)\beta(z)+(F(z)-F(x))E[Y\mid x<Y<z].$$
How do we maximize this?