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
  H is_point_conv_on X implies abs(H) is_point_conv_on X & lim (abs(H),X
  ) = |. lim (H,X) .| & -H is_point_conv_on X & lim (-H,X) = - lim(H,X)
proof
  assume
A1: H is_point_conv_on X;
A2: now
    let x;
    assume x in X;
    then H#x is convergent by A1,Th19;
    then abs(H#x) is convergent by SEQ_4:13;
    hence abs(H)#x is convergent by Th31;
  end;
A3: now
    let x;
    assume x in dom (-lim(H,X));
    then
A4: x in dom lim(H,X) by VALUED_1:8;
    then x in X by A1,Def13;
    then
A5: H#x is convergent by A1,Th19;
    thus (-lim(H,X)).x = -(lim(H,X).x) by VALUED_1:8
      .= -(lim(H#x)) by A1,A4,Def13
      .= lim(-(H#x)) by A5,SEQ_2:10
      .= lim((-H)#x) by Th31;
  end;
A6: X common_on_dom H by A1;
  then X common_on_dom abs(H) by Th37;
  hence
A7: abs(H) is_point_conv_on X by A2,Th19;
A8: now
    let x;
    assume x in dom |.lim(H,X).|;
    then
A9: x in dom lim(H,X) by VALUED_1:def 11;
    then x in X by A1,Def13;
    then
A10: H#x is convergent by A1,Th19;
    thus |.lim(H,X).|.x = |.lim(H,X).x.| by VALUED_1:18
      .= |.lim(H#x).| by A1,A9,Def13
      .= lim |.(H#x).| by A10,SEQ_4:14
      .= lim(abs(H)#x) by Th31;
  end;
A11: now
    let x;
    assume x in X;
    then H#x is convergent by A1,Th19;
    then -(H#x) is convergent by SEQ_2:9;
    hence (-H)#x is convergent by Th31;
  end;
  dom |.lim(H,X).| = dom lim(H,X) by VALUED_1:def 11
    .= X by A1,Def13;
  hence lim (abs(H),X) = |. lim (H,X) .| by A7,A8,Def13;
  X common_on_dom -H by A6,Th37;
  hence
A12: -H is_point_conv_on X by A11,Th19;
  dom (-lim(H,X)) = dom lim(H,X) by VALUED_1:8
    .= X by A1,Def13;
  hence thesis by A12,A3,Def13;
end;
