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

theorem
  for p,q being DTree-yielding FinSequence st x-tree p = y-tree q holds
  x = y & p = q
proof
  let p,q be DTree-yielding FinSequence;
  assume
A1: x-tree p = y-tree q;
 (x-tree p).{} = x by Def4;
  hence x = y by A1,Def4;
   dom (x-tree p) = tree(doms p) & dom (y-tree q) = tree(doms q) by Th10;
then A2: doms p = doms q by A1,TREES_3:50;
 dom p = dom doms p & dom doms q = dom q by TREES_3:37;
then A3: len p = len q by A2,FINSEQ_3:29;
 now
    let i be Nat;
    assume that
A4: i >= 1 and
A5: i <= len p;
    consider n be Nat such that
A6: i = 1+n by A4,NAT_1:10;
    reconsider n as Element of NAT by ORDINAL1:def 12;
A7: n < len p by A5,A6,NAT_1:13;
then  p.i = (x-tree p)|<*n*> by A6,Def4;
    hence p.i = q.i by A1,A3,A6,A7,Def4;
  end;
  hence thesis by A3;
end;
