In Freitag and Kiehl's etale cohomology book p206,$H(X_e,Rj_*(\Lambda|_{X_{\eta}}))\to H(X_{es},i^* Rj_*(\Lambda|_{X_{\eta}})$is a isomorphism,where X is a proper scheme and smooth over the base scheme S at all the point of Y, the complement of the scheme $X_e$.

If $X_e$ is proper,it is easily derived from the proper base change theorem.In this book,authors argue that,when F is locally constant,the $R\Gamma_Y F$ has the base change property,but I don't understand how they do this(The purity theorem is only proved in the case that X and Y are both smooth.)So how to prove this formula?