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
  H is_point_conv_on X
  implies r(#)H is_point_conv_on X & lim (r(#)H, X) = r(#)(lim(H, X))
  proof
    assume
    A1: H is_point_conv_on X;

    A3:
    now
      let x;
      assume
      A4: x in dom(r(#)(lim(H, X)));
      then
      A5: x in dom lim(H, X) by VFUNCT_1:def 4;
      then
      A6: x in X by A1, Def13;

      X1: lim(H, X)/.x = lim(H, X).x by PARTFUN1:def 6, A5;

      X2: (r(#)lim(H, X)).x = (r(#)lim(H, X))/.x by PARTFUN1:def 6, A4; 

      thus (r(#)lim(H, X)).x = r*(lim(H, X)/.x) by X2, A4, VFUNCT_1:def 4
      .= r*(lim(H#x)) by A1, A5, X1, Def13
      .= lim(r*(H#x)) by A6, A1, Th19, NORMSP_1:28
      .= lim((r(#)H)#x) by A1, A6, Th32;
    end;

    A8:
    now
      let x;
      assume
      A9: x in X;
      then r*(H#x) is convergent by A1, Th19, NORMSP_1:22;
      hence (r(#)H)#x is convergent by A1, A9, Th32;
    end;

    A10: r(#)H is_point_conv_on X by A1, Th38, A8, Th19;

    dom(r(#)(lim(H, X))) = dom lim(H, X) by VFUNCT_1:def 4
    .= X by A1, Def13;
    hence thesis by A10, A3, Def13;
  end;
