I'm trying to understand the proof at the top of page 3 of
http://math.stanford.edu/~vakil/02-245/sclass6A.pdf
http://math.stanford.edu/~vakil/02-245/sclass6B.pdf
Why can $D$ and D' be moved in their linear equivalence class in such a way that they intersect transversely AND such that their intersection lies in $U$? Intuitively this makes sense, of course, but what are the formal arguments?