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 p2 being FinSequence of FinTrees D2 st dom p2 = dom p &
  (for i st i in dom p ex T being Element of FinTrees [:D1,D2:] st
  T = p.i & p2.i = T`2) & ([d1,d2]-tree p)`2 = d2-tree p2
proof
  let D1,D2, d1,d2;
  let p be FinSequence of FinTrees [:D1,D2:];
  consider p2 being FinSequence of Trees D2 such that
  A1: dom
 p2 = dom p & for i st i in dom p ex T being Element of FinTrees [:D1,D2
  :] st T = p.i & p2.i = T`2 and
A2: ([d1,d2]-tree p)`2 = d2-tree p2 by Th30;
 rng p2 c= FinTrees D2
  proof
    let x be object;
    assume x in rng p2;
    then consider y being object such that
A3: y in dom p2 and
A4: x = p2.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: p2.y = T`2 by A1,A3;
 dom T`2 = dom T by Th24;
    hence thesis by A4,A5,TREES_3:def 8;
  end;
then  p2 is FinSequence of FinTrees D2 by FINSEQ_1:def 4;
  hence thesis by A1,A2;
end;
