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 Th20:
  H is_point_conv_on X implies (f = lim(H, X) iff dom f = X & for x
  st x in X holds for p st p>0 ex k st for n st n>=k
  holds ||.(H.n)/.x - f/.x.|| < p)
  proof
    assume
    A1: H is_point_conv_on X;
    thus f = lim(H, X) implies dom f = X & for x st x in X
    holds for p st p>0 ex k st for n st n>=k holds ||.(H.n)/.x - f/.x.|| < p
    proof
      assume
      A2: f = lim(H, X);
      hence
      A3: dom f = X by A1, Def13;
      let x;
      assume
      A4: x in X;
      then
      A5: H#x is convergent by A1, Th19;
      X6: f/.x = f.x by PARTFUN1:def 6, A4, A3;
      let p;
      assume
      A6: p>0;
      f.x = lim(H#x) by A1, A2, A3, A4, Def13;
      then consider k being Nat such that
      A7: for n being Nat st n>=k holds ||.(H#x).n - f/.x.|| < p
      by X6, A5, A6, NORMSP_1:def 7;
      take k;
      let n;
      assume n>=k;
      then ||.(H#x).n - f/.x.|| < p by A7;
      hence thesis by Def10;
    end;

    assume that
    A8: dom f = X and
    A9: for x st x in X holds for p st p>0 ex k st for n st n>=k holds ||.
    (H.n)/.x - f/.x.|| < p;
    now
      let x such that
      A10: x in dom f;
      X10: f/.x =f.x by A10, PARTFUN1:def 6;
      A11:
      now
        let p be Real;
        assume
         p > 0; then
        consider k such that
        A13: for n st n >= k holds ||.(H.n)/.x - f/.x.|| < p by A8, A9, A10;
        take k;
        let n be Nat;
        assume n >= k;
        then ||.(H.n)/.x - f/.x.|| < p by A13;
        hence ||.(H#x).n - f/.x.|| < p by Def10;
      end;
      then H#x is convergent by NORMSP_1:def 6;
      hence f.x = lim(H#x) by X10, A11, NORMSP_1:def 7;
    end;
    hence thesis by A1, A8, Def13;
  end;
