reserve x, y, z for object,
  i, j, n for Nat,
  p, q, r for FinSequence,
  v for FinSequence of NAT;

theorem
  root-tree x = y-tree p & p is DTree-yielding implies x = y & p = {}
proof
  assume that
A1: root-tree x = y-tree p and
A2: p is DTree-yielding;
  reconsider p9 = p as DTree-yielding FinSequence by A2;
  thus x = (root-tree x).{} by Th3
    .= y by A1,A2,Def4;
 dom (y-tree p) = tree(doms p9) by Th10;
then A3: doms p9 = {} by A1,TREES_3:50,52;
 dom doms p9 = dom p by TREES_3:37;
  hence thesis by A3;
end;
