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. Since-q-ix-i,-?-i, S i ) is winning we get that (q i , r ) ? ? i (x i )(S i ) for some r . Moreover, as x i is a variable free in N , we either have ? i (x i ) = ? N (x i ) (when N is of the form x i N 1 . . . N p ) or ? i (x i ) = ? N (x i ) r for some r

. So, there is r i such that (q i , r i ) ? ? N (x i )(S i ) But then, as we have noted above, Eve has a winning strategy from q : (term(t i ), ?, S i ) is