- occurred. Consider a query $x_0, x_1, \ldots, x_{n-1}$. If it's the
- same as an earlier query, then $A$ learns nothing (because it could
- have remembered the answer from last time). If it's \emph{not} a
- prefix of some previous query, then we must add a new edge to our
- graph; then either the bad event occurs or we create a new node for
- the result, and because $F$ is a random function, the answer is
- uniformly random. Finally, we consider the case where the query is a
- prefix of some earlier query, or queries. But these were computed at
- random at the time.
+ occurred. Consider a query $x_0, x_1, \ldots, x_{n-1}$. If it's the same
+ as an earlier query, then $A$ learns nothing (because it could have
+ remembered the answer from last time). If it's a \emph{prefix} of some
+ earlier query, then the answer is the value of some internal node which
+ hasn't been revealed before; however, the value of that internal node was
+ chosen uniformly at random (we claim). Finally, if the query is not a
+ prefix of any previous query, then we add a new edge to our graph. If the
+ bad event doesn't occur, we must add a new node for the result, and the
+ value at that node will be uniformly random, because $F$ is a random
+ function being evaluated at a new point -- this is the only time we add new
+ nodes to the graph, justifying the claim made earlier.