reserve x, y, z for object,
  i, j, n for Nat,
  p, q, r for FinSequence,
  v for FinSequence of NAT;
reserve T,T9 for DecoratedTree,
  x,y for set;
reserve D1, D2 for non empty set,
  T for DecoratedTree of D1,D2,
  d1 for Element of D1,
  d2 for Element of D2,
  F for non empty DTree-set of D1,D2,
  F1 for non empty (DTree-set of D1),
  F2 for non empty DTree-set of D2;

theorem Th25:
  (root-tree [d1,d2])`1 = root-tree d1 & (root-tree [d1,d2])`2 = root-tree d2
proof reconsider r = {} as Node of root-tree [d1,d2] by TREES_1:22;
A1: dom (root-tree [d1,d2])`1 = dom root-tree [d1,d2] by Th24;
A2: dom (root-tree [d1,d2])`2 = dom root-tree [d1,d2] by Th24;
A3: (root-tree [d1,d2]).r = [d1,d2];
A4: [d1,d2]`1 = d1;
A5: [d1,d2]`2 = d2;
  thus (root-tree [d1,d2])`1 = root-tree ((root-tree [d1,d2])`1.r) by A1,Th5
    .= root-tree d1 by A3,A4,TREES_3:39;
  thus (root-tree [d1,d2])`2 = root-tree ((root-tree [d1,d2])`2.r) by A2,Th5
    .= root-tree d2 by A3,A5,TREES_3:39;
end;
