theorem
  for T1, T2 being DecoratedTree holds
  dom (x-tree (T1,T2)) = tree(dom T1, dom T2)
proof
  let T1, T2 be DecoratedTree;
   dom
 (x-tree <*T1,T2*>) = tree(doms <*T1,T2*>) & doms <*T1,T2*> = <*dom T1,
  dom T2*> by Th10,FINSEQ_3:133;
  hence thesis by TREES_3:def 17;
end;
