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_unif_conv_on X implies H is_unif_conv_on Y
proof
  assume that
A1: Y c= X and
A2: Y<>{} and
A3: H is_unif_conv_on X;
  consider f such that
A4: dom f = X and
A5: for p st p>0 ex k st for n,x st n>=k & x in X holds |.(H.n).x - f.
  x.| < p by A3;
A6: now
    take g = f|Y;
    thus
A7: dom g = dom f /\ Y by RELAT_1:61
      .= Y by A1,A4,XBOOLE_1:28;
    let p;
    assume p>0;
    then consider k such that
A8: for n,x st n>=k & x in X holds |.(H.n).x - f.x.| < p by A5;
    take k;
    let n,x;
    assume that
A9: n>=k and
A10: x in Y;
    |.(H.n).x - f.x.| < p by A1,A8,A9,A10;
    hence |.(H.n).x - g.x.| < p by A7,A10,FUNCT_1:47;
  end;
  X common_on_dom H by A3;
  then Y common_on_dom H by A1,A2,Th22;
  hence thesis by A6;
end;
