DNA test to prove that two grand-grand-children descend from same grand-grandparents Can DNA test prove that two persons descend from same great-grandparents (or test where one great-grandchild and grand-child are tested)?

Yes testing for relatedness of grand-grand children is possible. (Ignoring some very unlikely, but theoretically possible cases.)

Grand-grand children are still highly related genetically, and this degree of relation is also shown in standard consumer grade commercial DNA testing kits (such as 23andme, where you could link to other people and see their degree of relatedness to you – even further beyond grand-grand children - or where you could even search for relatives, whom you may not know personally).