theorem Th38:
  X common_on_dom H implies X common_on_dom r(#)H
proof
  assume
A1: X common_on_dom H;
A2: now
    let n;
    dom (H.n) = dom(r(#)(H.n)) by VALUED_1:def 5
      .= dom ((r(#)H).n) by Def1;
    hence X c= dom ((r(#)H).n) by A1;
  end;
  X <> {} by A1;
  hence thesis by A2;
end;
