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;

theorem
  Z c= X & Z<>{} & H is_point_conv_on X implies H is_point_conv_on Z &
  lim(H, X)|Z = lim(H,Z)
  proof
    assume that
    A1: Z c= X and
    A2: Z<>{} and
    A3: H is_point_conv_on X;
    consider f such that
    A4: X = dom f and
    A5: for x st x in X holds for p st p>0 ex k st for n st n>=k 
    holds ||.(H.n)/.x - f/.x.|| < p by A3;

    now
      take g = f|Z;
      thus
      A7: Z = dom g by A1, A4, RELAT_1:62;
      let x;
      assume
      A8: x in Z;
      then
      X8: f/.x =f.x by A1, A4, PARTFUN1:def 6;

      g/.x =g.x by A7, A8, PARTFUN1:def 6;
      then     
      X10: g/.x = f/.x by A7, A8, FUNCT_1:47, X8;

      let p;
      assume p>0;
      then consider k such that
      A9: for n st n>=k holds ||.(H.n)/.x - f/.x.|| < p by A1, A5, A8;
      take k;

      let n;
      assume n>=k;
      hence ||.(H.n)/.x -g/.x.|| < p by A9, X10;
    end;
    hence
    A10: H is_point_conv_on Z by A1, A2, A3, Th22;

    A11:
    now
      let x;
      assume
      A12: x in dom (lim(H, X)|Z);
      then
      A13: x in (dom lim(H, X)) /\ Z by RELAT_1:61;
      then
      A14: x in dom lim(H, X) by XBOOLE_0:def 4;
      x in Z by A13, XBOOLE_0:def 4;
      then
      A15: x in dom lim(H,Z) by A10, Def13;
      thus (lim(H, X)|Z).x = (lim(H, X)).x by A12, FUNCT_1:47
      .= lim (H#x) by A3, A14, Def13
      .= (lim(H,Z)).x by A10, A15, Def13;
    end;

    dom lim(H, X) = X by A3, Def13;
    then dom lim(H, X) /\ Z = Z by A1, XBOOLE_1:28;
    then dom (lim(H, X)|Z) = Z by RELAT_1:61;
    then dom (lim(H, X)|Z) = dom lim(H,Z) by A10, Def13;
    hence thesis by A11, PARTFUN1:5;
  end;
