reserve x for set,
  t,t1,t2 for DecoratedTree;
reserve C for set;
reserve X,Y for non empty constituted-DTrees set;

theorem Th34:
  for D being non empty set, r being FinSequence of D, r1 being
FinSequence st 1 <= len r & r1 = r|Seg 1 holds ex x being Element of D st r1 =
  <*x*>
proof
  let D be non empty set, r be FinSequence of D, r1 be FinSequence;
  assume that
A1: 1 <= len r and
A2: r1 = r|Seg 1;
  consider x being set such that
A3: x = r1.1;
  1 in {1} by TARSKI:def 1;
  then 1 in dom r1 by A1,A2,FINSEQ_1:2,17;
  then
A4: x in rng r1 by A3,FUNCT_1:def 3;
  len r1 = 1 by A1,A2,FINSEQ_1:17;
  then
A5: r1 = <*x*> by A3,FINSEQ_1:40;
  r1 is_a_prefix_of r by A2,TREES_1:def 1;
  then ex q1 being FinSequence st r = r1^q1 by TREES_1:1;
  then reconsider r9 = r1 as FinSequence of D by FINSEQ_1:36;
  rng r9 c= D;
  hence thesis by A5,A4;
end;
