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 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;
