theorem Th21:
  H is_unif_conv_on X implies H is_point_conv_on X
  proof
    assume
    A1: H is_unif_conv_on X;

    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;
    hence thesis by A1;
  end;
