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 Th19:
  H is_point_conv_on X iff X common_on_dom H & for x st x in X
  holds (H#x) is convergent
proof
  defpred X[set] means $1 in X;
  deffunc U(Element of D) = In(lim(H#$1),REAL);
  consider f such that
A1: for x holds x in dom f iff X[x] and
A2: for x st x in dom f holds f.x = U(x) from SEQ_1:sch 3;
  thus H is_point_conv_on X implies X common_on_dom H & for x st x in X holds
  (H#x) is convergent
  proof
    assume
A3: H is_point_conv_on X;
    hence X common_on_dom H;
    let x;
    assume
A4: x in X;
    ex f st X = dom f & for x st x in X holds (H#x) is convergent & lim(H#
    x) = f.x by A3,Th18;
    hence thesis by A4;
  end;
  assume that
A5: X common_on_dom H and
A6: for x st x in X holds (H#x) is convergent;
  now
    take f;
    thus
A7: X = dom f
    proof
      thus X c= dom f
      proof
        let x be object such that
A8:     x in X;
        X c= dom (H.0) by A5;
        then X c= D by XBOOLE_1:1;
        then reconsider x9 = x as Element of D by A8;
        x9 in dom f by A1,A8;
        hence thesis;
      end;
      let x be object;
      assume x in dom f;
      hence thesis by A1;
    end;
    let x;
    assume
A9: x in X;
    hence (H#x) is convergent by A6;
    thus f.x = U(x) by A2,A7,A9
       .= lim(H#x);
  end;
  hence thesis by A5,Th18;
end;
