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They can still travel without a ID if it was lost or stolen. They need to inform the airline that the passport was stolen. They can also offer a copy of the police report to help. Also they should show the picture they took. If they have credit cards, they should also present those as well. A copy of their birth certificate (in English) would also be helpeful, if available.
At security, they also will not require an ID but they will be subjected to a more thorough security screening, where the officer might search their carry on by hand and they may be subjected to a pat-down. This may not happen, since presumably the machines are meant to find anything. But it is possible that a pat-down and hand search occurs, since she does not have an ID.
Overall, she should be able to fly even without an ID. Here,is information on that: http://www.tsa.gov/traveler-information/acceptable-ids%20. This also includes a list of items they can try to bring that would help in identification.
Here is additional information about traveling without an ID, on the TSA Blog: http://blog.tsa.gov/2013/04/tsa-travel-tips-tuesday-can-you-fly.html
And here is a list of less common items that can be used as IDs for traveling: http://thepointsguy.com/2014/09/12-crazy-forms-of-id-to-get-through-airport-security/, including cell phone pictures of the ID.
All of this should help the traveler get through security and on the airplane for travel.
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