
theorem Th1:
  for t being DecoratedTree holds t|<*>NAT = t
proof
  let t be DecoratedTree;
A1: dom (t|<*>NAT) = (dom t)|<*>NAT by TREES_2:def 10;
  now
    let p be FinSequence of NAT;
    assume p in dom (t|<*>NAT);
    hence (t|<*>NAT).p = t.({}^p) by A1,TREES_2:def 10
      .= t.p by FINSEQ_1:34;
  end;
  hence thesis by A1,TREES_1:31;
end;
