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 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;
    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 A5,A6,SEQ_2:def 7;
     reconsider k as Nat;
    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;
A11: now
      let p be Real;
      assume
A12:  p>0;
      consider k such that
A13:  for n st n>=k holds |.(H.n).x - f.x.| < p by A8,A9,A10,A12;
       reconsider k as Nat;
      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 SEQ_2:def 6;
    hence f.x = lim(H#x) by A11,SEQ_2:def 7;
  end;
  hence thesis by A1,A8,Def13;
end;
