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 Th18:
  H is_point_conv_on X iff X common_on_dom H & ex f st X = dom f &
  for x st x in X holds (H#x) is convergent & lim(H#x) = f.x
  proof
    thus H is_point_conv_on X implies X common_on_dom H & ex f st X = dom f &
    for x st x in X holds (H#x) is convergent & lim(H#x) = f.x
    proof
      assume
      A1: H is_point_conv_on X;
      hence X common_on_dom H;
      consider f such that
      A2: X = dom f and
      A3: 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 by A1;
      take f;

      thus X = dom f by A2;

      let x;
      assume
      A4: x in X;
      then
      X4: f/.x = f.x by A2, PARTFUN1:def 6;

      A5:
      now
        let p be Real;
        assume
        A6: p>0;
        consider k such that
        A7: for n st n >= k holds ||.(H.n)/.x - f/.x.|| < p by A3, A4, A6;
        take k;

        let n be Nat;
        assume n >= k;
        then ||.(H.n)/.x - f/.x.|| < p by A7;
        hence ||.(H#x).n - f/.x.|| < p by Def10;
      end;
      hence (H#x) is convergent by NORMSP_1:def 6;
      hence thesis by X4, A5, NORMSP_1:def 7;
    end;

    assume
    A8: X common_on_dom H;

    given f such that
    A9: X = dom f and
    A10: for x st x in X holds (H#x) is convergent & lim(H#x) = f.x;

    ex f st X = dom f
    & 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
      take f;
      thus X = dom f by A9;
      let x;
      assume X10: x in X;
      then
      A11: (H#x) is convergent & lim(H#x) = f.x by A10;
      X11: f/.x =f.x by PARTFUN1:def 6, X10, A9;
      let p;
      assume p>0;
      then consider k being Nat such that
      A12: for n being Nat st n >= k holds ||.(H#x).n - f/.x.|| < p
      by A11, NORMSP_1:def 7, X11;
      take k;
      let n;
      assume n >= k;
      then ||.(H#x).n - f/.x.|| < p by A12;
      hence thesis by Def10;
    end;
    hence thesis by A8;
  end;
