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
  for D1,D2, d1,d2 for p being FinSequence of FinTrees [:D1,D2:]
  ex p1 being FinSequence of FinTrees D1 st dom p1 = dom p &
  (for i st i in dom p ex T being Element of FinTrees [:D1,D2:] st
  T = p.i & p1.i = T`1) & ([d1,d2]-tree p)`1 = d1-tree p1
proof
  let D1,D2, d1,d2;
  let p be FinSequence of FinTrees [:D1,D2:];
  consider p1 being FinSequence of Trees D1 such that
  A1: dom
 p1 = dom p & for i st i in dom p ex T being Element of FinTrees [:D1,D2
  :] st T = p.i & p1.i = T`1 and
A2: ([d1,d2]-tree p)`1 = d1-tree p1 by Th29;
 rng p1 c= FinTrees D1
  proof
    let x be object;
    assume x in rng p1;
    then consider y being object such that
A3: y in dom p1 and
A4: x = p1.y by FUNCT_1:def 3;
    reconsider y as Nat by A3;
    consider T being Element of FinTrees [:D1,D2:] such that
    T = p.y and
A5: p1.y = T`1 by A1,A3;
 dom T`1 = dom T by Th24;
    hence thesis by A4,A5,TREES_3:def 8;
  end;
then  p1 is FinSequence of FinTrees D1 by FINSEQ_1:def 4;
  hence thesis by A1,A2;
end;
