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;
reserve x for Element of D;

theorem Th42:
  H is_unif_conv_on X
  iff X common_on_dom H & H is_point_conv_on X
  & for r st 0<r ex k st for n, x st n >= k & x in X
  holds ||.(H.n)/.x-(lim(H, X))/.x.|| < r
  proof
    thus H is_unif_conv_on X implies
    X common_on_dom H & H is_point_conv_on X
    & for r st 0<r ex k st for n,x st n>=k & x in X
    holds ||.(H.n)/.x-(lim(H, X))/.x.||<r
    proof
      assume
      A1: H is_unif_conv_on X;
      then consider f such that
      A2: X = dom f and
      A3: for p st p>0 ex k st for n,x st n>= k & x in X
      holds ||.(H.n)/.x-f/.x .|| < p;

      thus X common_on_dom H by A1;

      A4:
      now
        let x such that
        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 A3;
        take k;

        let n;
        assume n >= k;
        hence ||.(H.n)/.x-f/.x.||<p by A5, A6;
      end;

      thus H is_point_conv_on X by A1, Th21;
      then
      A7: f = (lim(H, X)) by A2, A4, Th20;

      let r;
      assume r > 0;
      then consider k such that
      A8: for n,x st n>=k & x in X holds ||.(H.n)/.x-f/.x.||<r by A3;
      take k;

      let n,x;
      assume n >= k & x in X;
      hence thesis by A7, A8;
    end;
    assume that
    A9: X common_on_dom H and
    A10: H is_point_conv_on X and
    A11: for r st 0 < r ex k st for n, x st n >= k & x in X
    holds ||.(H.n)/.x-(lim(H, X))/.x.|| < r;
    dom lim(H, X) = X by A10, Def13;
    hence thesis by A9, A11;
  end;
