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 Th29:
  for D1,D2, d1,d2, F for p being FinSequence of F
  ex p1 being FinSequence of Trees D1 st dom p1 = dom p &
  (for i st i in dom p ex T being Element of F st T = p.i & p1.i = T`1) &
  ([d1,d2]-tree p)`1 = d1-tree p1
proof
  let D1,D2, d1,d2, F;
  let p be FinSequence of F;
A1: Seg len p = dom p by FINSEQ_1:def 3;
  defpred X[set,set] means ex T being Element of F st T = p.$1 & $2 = T`1;
A2: for i be Nat st i in Seg len p ex x being Element of Trees D1 st X[i,x]
  proof
    let i be Nat;
    assume i in Seg len p;
    then reconsider T = p.i as Element of F by A1,Lm1;
    reconsider y = T`1 as Element of Trees D1 by TREES_3:def 7;
    take y, T;
    thus thesis;
  end;
  consider p1 being FinSequence of Trees D1 such that
A3: dom p1 = Seg len p & for i be Nat st i in Seg len p holds X[i,p1.i]
  from FINSEQ_1:sch 5(A2);
  take p1;
  thus
A4: dom p1 = dom p by A3,FINSEQ_1:def 3;
  hence for i st i in dom p
  ex T being Element of F st T = p.i & p1.i = T`1 by A3;
 now
    let i;
    assume i in dom p;
then  ex T being Element of F st T = p.i & p1.i = T`1 by A3,A4;
    hence for T st T = p.i holds p1.i = T`1;
  end;
  hence thesis by A4,Th27;
end;
