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

theorem
  x-flat_tree({}) = root-tree x & x-tree({}) = root-tree x
proof
 len {} = 0;
then A1: dom (x-flat_tree {}) = elementary_tree 0 by Def3;
 now
    let y be object;
    assume y in elementary_tree 0;
then  y = {} by TARSKI:def 1,TREES_1:29;
    hence (x-flat_tree {}).y = x by Def3;
  end;
  hence x-flat_tree({}) = root-tree x by A1;
  reconsider e = {} as DTree-yielding FinSequence;
A2: dom (x-tree {}) = tree(doms e) by Th10
    .= elementary_tree 0 by FUNCT_6:23,TREES_3:52;
 now
    let y be object;
    assume y in elementary_tree 0;
then  y = {} by TARSKI:def 1,TREES_1:29;
    hence (x-tree e).y = x by Def4;
  end;
  hence thesis by A2;
end;
