deletions | additions       diff --git a/We_start_by_applying_the__.tex b/We_start_by_applying_the__.tex  index f89e90d..a2461cf 100644  --- a/We_start_by_applying_the__.tex  +++ b/We_start_by_applying_the__.tex 
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We split the previous expression as   \begin{multline}  I\propto (2\pi)^2 \Biggr\{   \int |t_{1}|^2 \Biggr\{ [1-f(\epsilon_j-\mu)] P_{1}(\mu_d-\epsilon_j) d\epsilon_j\\  \int |t_{2}|^2 f(\epsilon_i-\mu) P_{2}(\epsilon_i-\mu_d) d\epsilon_i   -\\   \int |t_{1}|^2 [1-f(\epsilon_j-\mu)] P_{2}(\mu_d-\epsilon_j) d\epsilon_j\\  \int |t_{2}|^2 f(\epsilon_i-\mu) P_{1}(\epsilon_i-\mu_d) d\epsilon_i   \Biggr\}  \end{multline} 
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d\epsilon_i d\epsilon_j  \end{multline}