theorem Th37:
  X common_on_dom H implies X common_on_dom ||.H.|| & X common_on_dom (-H)
  proof
    assume
    A1: X common_on_dom H;

    now
      let n;
      dom (H.n) = dom ||.H.n.|| by NORMSP_0:def 3
      .= dom (||.H.||.n) by Def4;
      hence X c= dom (||.H.||.n) by A1;
    end;
    hence X common_on_dom ||.H.|| by A1;

    now
      let n;
      dom (H.n) = dom (-(H.n)) by VFUNCT_1:def 5
      .= dom ((-H).n) by Def3;
      hence X c= dom ((-H).n) by A1;
    end;
    hence thesis by A1;
  end;
