reserve D for non empty set,
  D1, D2, x, y, Z for set,
  n, k for Nat,
  p, x1, r for Real,
  f for Function,
  Y for RealNormSpace,
  G, H, H1, H2, J for Functional_Sequence of D,the carrier of Y;
reserve
  x for Element of D,
  X for set,
  S1, S2 for sequence of Y,
  f for PartFunc of D,the carrier of Y;
reserve x for Element of D;

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;
