At the first stage where the observed event is true, adjoin it as a distinguished root to the execution history. The resulting proper witness tree has no further copy of the root and all of its other labels lie in $Gamma(B)$. The witness-tree lemma bounds the probability of every such tree, and the branching-process sum is exactly the local-lemma neighborhood product. The output event is a subset of the event that the observation was true at some stage.