I don't know the software you are using, as I use a dif kind of video editor.
But the video editing process is based in small pieces - video clips - that you put togheter in order to produce a movie file.
So, when you cut to take out the 15secs, you should:
1. - cut a piece from the start till the point the unwanted part begins - you get the 1st piece;
2. - take the 15secs from the other part of movie (till the end) - you get the second piece;
3. - put piece 1 and 2 togheter and connect them via a transitoin or fading;
4. - encode the movie and produce the final result you want.
However, take into consideration that if you start with a DVD video you can only edit if you import (transform it to) and get a Mpeg2 or other compatible video file that your editor can handle.