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;

theorem
  T.p = (T|p).{}
proof
  <*>NAT in (dom T)|p by TREES_1:22;
  then (T|p).<*>NAT = T.(p^<*>NAT) by TREES_2:def 10
    .= T.p by FINSEQ_1:34;
  hence thesis;
end;
