reserve x for set,
  t,t1,t2 for DecoratedTree;
reserve C for set;
reserve X,Y for non empty constituted-DTrees set;
reserve T for DecoratedTree,
  p for FinSequence of NAT;
reserve T for finite-branching DecoratedTree,
  t for Element of dom T,
  x for FinSequence,
  n, m for Nat;

theorem Th39:
  t^<*n*> in dom T iff n+1 in dom (t succ)
proof
  now
    assume t^<*n*> in dom T;
    then t^<*n*> in succ t;
    then n < card succ t by Th7;
    then n < len (t succ) by Def5;
    then
A1: n+1 <= len (t succ) by NAT_1:13;
    0+1 <= n+1 by XREAL_1:6;
    then n+1 in Seg len (t succ) by A1,FINSEQ_1:1;
    hence n+1 in dom (t succ) by FINSEQ_1:def 3;
  end;
  hence t^<*n*> in dom T implies n+1 in dom (t succ);
  assume n+1 in dom (t succ);
  then n+1 in Seg len (t succ) by FINSEQ_1:def 3;
  then n+1 <= len (t succ) by FINSEQ_1:1;
  then n < len (t succ) by NAT_1:13;
  then n < card succ t by Def5;
  then t^<*n*> in succ t by Th7;
  hence thesis;
end;
