Are roots a danger, how to remove them

A neighbor moved into her townhouse 20 years ago and there were no roots in her backyard.
Now they have a beautiful yard with perfect grass and benches facing the woods and other nice stuff, but roots have reached into the yard, one 40 feet from where it surfaces, which is about 10 feet from the closest tree, all of which are outside their property.
There are 4 other root which have gone 20 feet.
The part of the roots that show start off neear the trees as about 3 inches wide, but most of them is under the grournd. They must be 6 inches in diameter.
The trees are growing in the stream bed and the base of the trees are 2 to 4 feet below the level of their lawn. That means the roots start out that deep but rise to the surface within 10 feet of the tree (assuming it's the closest tree. The sun was in my eyes and I haven't identified any of the trees.)
A She would like to get rid of them because the yard would be nicer.
B. She is worried that they are heading for her basement, one that is 6.5 feet below grade, with a one fooot high window at ground level, It's made with cinder blocks.
This is the big, more important question? Are the roots going to damage her foundation?
I suggested cutting a 6" gap in the big roots which would stop them from growing, but she said something else bad would happen and she told me what it was, but the phone was noisy and I missed it. Is she right?
Problem A, again. As to getting rid of the roots, would they be hard to dig up? I assume that even though sub-roots aren't coming out of the top, they are coming out of the bottom and holding the bigger root in place, and that every foot will be a struggle.
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On 5/11/2015 7:33 PM, micky wrote:

I'd cut them off myself. Since trees are not even on her property she can't harm the trees. I've done it to my own trees.
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On 05/11/2015 07:33 PM, micky wrote:

I've seen the damage tree roots have done to foundations and I've seen the damage caused by storms so yes, fire up the Stihl and take that bad boy down.
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clipped

I would do nothing to the trees without expert advice, because:
If the trees drink a lot, they might help keep a balance of groundwater in yard.
Cutting roots often prompts plant to grow more! It is called root pruning.
Surface roots, if injured (by mowers, etc.) will grow a new stem/plant from the injured spot.
That said, the variety, height and distance from building would help in estimating danger to the basement.
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