reserve D for non empty set,
  D1,D2,x,y for set,
  n,k for Nat,
  p,x1 ,r for Real,
  f for Function;
reserve F for Functional_Sequence of D1,D2;
reserve G,H,H1,H2,J for Functional_Sequence of D,REAL;
reserve x for Element of D,
  X,Y for set,
  S1,S2 for Real_Sequence,
  f for PartFunc of D,REAL;

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;
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;
       reconsider k as Nat;
      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 SEQ_2:def 6;
    hence thesis by A5,SEQ_2: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 x in X;
    then
A11: (H#x) is convergent & lim(H#x) = f.x by A10;
    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,SEQ_2:def 7;
     reconsider k as Nat;
    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;
