A proposition from Neyman-Pearson lemma

The following proposition stems from the proof of Neyman-Pearson lemma. The result is not quite intuitive, and is there any easier proof?

Proposition 1 Let {H, F} are two random variables. Consider {V = \inf_{a\ge 0} \{x a + \mathbb{E}[ (H - a F)^+]\}}. Suppose there exists {\hat a} satisfying

\displaystyle \mathbb{E}[F I_{\{\hat a F<H\}}] = x,

then {V = x \hat a + \mathbb{E}[(H- \hat a F)^+] = \mathbb{P}\{\hat a F<H\}}.

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