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 Th21:
  H is_unif_conv_on X implies H is_point_conv_on X
proof
  assume
A1: H is_unif_conv_on X;
A2: now
    consider f such that
A3: X = dom f and
A4: 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 A1;
    take f;
    thus X = dom f by A3;
    let x;
    assume
A5: x in X;
    let p;
    assume p>0;
    then consider k such that
A6: for n,x st n>=k & x in X holds |.(H.n).x - f.x.| < p by A4;
    take k;
    let n;
    assume n>=k;
    hence |.(H.n).x - f.x.| < p by A5,A6;
  end;
  X common_on_dom H by A1;
  hence thesis by A2;
end;
