YouTube Video Posting Question

I uploaded an interview that my production company did with the Obama Girl back in April to YouTube. During the 8 minute interview, we showed some of the "Crush on Obama" video and we had it playing on a screen during the interview. We also put up a lower third displaying the website while the video was playing. So in total, you could only see, not hear, the video for a total of 3 1/2 minutes of the 8 minutes.

I emailed the owner of the company to let him know what I was doing and gave him the link. In a matter of minutes I received an email from YouTube stating that "Next New Networks has claimed some or all visual content in your video...Your video is still live because Next New Networks has authorized the use of this content on YouTube. As long as Next New Networks has a claim on your video, they will receive public statistics about your video, such as number of views."

I understand that yes I did use their stuff during the interview.

My question is how do I do it in the future where I can claim it soley as ours? I noticed in their video that they have pics of Obama. Did they have to pay for those pics in order to use them and claim as theirs?

What's the process?

TIA

Sam