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

theorem
  for t being finite DecoratedTree, p being Element of dom t holds len
  succ(t,p) = len (p succ) & dom succ(t,p) = dom (p succ)
proof
  let t be finite DecoratedTree, p be Element of dom t;
  ( ex q being Element of dom t st q = p & succ(t,p) = t*(q succ))& rng (p
  succ) c= dom t by Def6;
  then dom succ(t,p) = dom (p succ) by RELAT_1:27;
  hence thesis by FINSEQ_3:29;
end;
