theorem Th38:
  X common_on_dom H implies X common_on_dom r(#)H
  proof
    assume
    A1: X common_on_dom H;
    now
      let n;
      dom (H.n) = dom(r(#)(H.n)) by VFUNCT_1:def 4
      .= dom ((r(#)H).n) by Def1;
      hence X c= dom ((r(#)H).n) by A1;
    end;
    hence thesis by A1;
  end;
