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;
reserve x, x9 for Element of dom T,
  y9 for set;

theorem
  x9 in succ x implies T.x9 in rng succ(T,x)
proof
  assume x9 in succ x;
  then consider n such that
A1: x9 = x^<*n*> and
  x^<*n*> in dom T;
A2: T.x9 = (succ(T,x)).(n+1) by A1,Th40;
  dom (succ(T,x)) = dom (T*(x succ)) by Th36
    .= dom (x succ) by Th37;
  then n+1 in dom succ(T,x) by A1,Th39;
  hence thesis by A2,FUNCT_1:def 3;
end;
