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 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;
