For states $|\alphaangle$ and $|\betaangle$, define time-reversed states by $|\tilde{\alpha}angle=\Theta|\alphaangle$ and $|\tilde{\beta}angle=\Theta|\betaangle$. Show that the overlap of the time-reversed states is given by $\langle\tilde{\beta} \mid \tilde{\alpha}angle=\langle\beta \mid \alphaangle^{*}$. Expand $|\tilde{\alpha}angle$ and $|\tilde{\beta}angle$ in complete sets of states using $\sum_{a}|aangle\langle a|=1$, and use that the complex conjugation operator has no effect on a ket, $\mathscr{K}|\alphaangle=|\alphaangle$.