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

theorem
  Y c= X & Y<>{} & H is_point_conv_on X implies H is_point_conv_on Y &
  lim(H,X)|Y = lim(H,Y)
proof
  assume that
A1: Y c= X and
A2: Y<>{} 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;
A6: now
    take g = f|Y;
    thus
A7: Y = dom g by A1,A4,RELAT_1:62;
    let x;
    assume
A8: x in Y;
    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;
    then |.(H.n).x - f.x.| < p by A9;
    hence |.(H.n).x - g.x.| < p by A7,A8,FUNCT_1:47;
  end;
  X common_on_dom H by A3;
  then Y common_on_dom H by A1,A2,Th22;
  hence
A10: H is_point_conv_on Y by A6;
A11: now
    let x;
    assume
A12: x in dom (lim(H,X)|Y);
    then
A13: x in (dom lim(H,X)) /\ Y by RELAT_1:61;
    then
A14: x in dom lim(H,X) by XBOOLE_0:def 4;
    x in Y by A13,XBOOLE_0:def 4;
    then
A15: x in dom lim(H,Y) by A10,Def13;
    thus (lim(H,X)|Y).x = (lim(H,X)).x by A12,FUNCT_1:47
      .= lim (H#x) by A3,A14,Def13
      .= (lim(H,Y)).x by A10,A15,Def13;
  end;
  dom lim(H,X) = X by A3,Def13;
  then dom lim(H,X) /\ Y = Y by A1,XBOOLE_1:28;
  then dom (lim(H,X)|Y) = Y by RELAT_1:61;
  then dom (lim(H,X)|Y) = dom lim(H,Y) by A10,Def13;
  hence thesis by A11,PARTFUN1:5;
end;
