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
  H1 is_point_conv_on X & H2 is_point_conv_on X implies H1+H2
  is_point_conv_on X & lim(H1+H2,X) = lim(H1,X) + lim(H2,X) & H1-H2
  is_point_conv_on X & lim(H1-H2,X) = lim(H1,X) - lim(H2,X) & H1(#)H2
  is_point_conv_on X & lim(H1(#)H2,X) = lim(H1,X) (#) lim(H2,X)
proof
  assume that
A1: H1 is_point_conv_on X and
A2: H2 is_point_conv_on X;
A3: now
    let x;
    assume
A4: x in X;
    then H1#x is convergent & H2#x is convergent by A1,A2,Th19;
    then (H1#x)+(H2#x) is convergent by SEQ_2:5;
    hence (H1+H2)#x is convergent by A1,A2,A4,Th33;
  end;
A5: now
    let x;
    assume
A6: x in X;
    then H1#x is convergent & H2#x is convergent by A1,A2,Th19;
    then (H1#x)-(H2#x) is convergent by SEQ_2:11;
    hence (H1-H2)#x is convergent by A1,A2,A6,Th33;
  end;
A7: X common_on_dom H1 & X common_on_dom H2 by A1,A2;
  then X common_on_dom H1+H2 by Th36;
  hence
A8: H1+H2 is_point_conv_on X by A3,Th19;
A9: now
    let x;
    assume
A10: x in dom (lim(H1,X)+lim(H2,X));
    then
A11: x in (dom lim(H1,X))/\(dom lim(H2,X)) by VALUED_1:def 1;
    then
A12: x in (dom lim(H2,X)) by XBOOLE_0:def 4;
A13: x in (dom lim(H1,X)) by A11,XBOOLE_0:def 4;
    then
A14: x in X by A1,Def13;
    then
A15: H1#x is convergent & H2#x is convergent by A1,A2,Th19;
    thus (lim(H1,X) + lim(H2,X)).x = (lim(H1,X)).x + (lim(H2,X)).x by A10,
VALUED_1:def 1
      .= lim(H1#x) + (lim(H2,X)).x by A1,A13,Def13
      .= lim(H1#x) + lim(H2#x) by A2,A12,Def13
      .= lim((H1#x) + (H2#x)) by A15,SEQ_2:6
      .= lim((H1+H2)#x) by A1,A2,A14,Th33;
  end;
A16: now
    let x;
    assume
A17: x in X;
    then H1#x is convergent & H2#x is convergent by A1,A2,Th19;
    then (H1#x)(#)(H2#x) is convergent by SEQ_2:14;
    hence (H1(#)H2)#x is convergent by A1,A2,A17,Th33;
  end;
A18: now
    let x;
    assume x in dom (lim(H1,X)(#)lim(H2,X));
    then
A19: x in (dom lim(H1,X))/\(dom lim(H2,X)) by VALUED_1:def 4;
    then
A20: x in (dom lim(H2,X)) by XBOOLE_0:def 4;
A21: x in (dom lim(H1,X)) by A19,XBOOLE_0:def 4;
    then
A22: x in X by A1,Def13;
    then
A23: H1#x is convergent & H2#x is convergent by A1,A2,Th19;
    thus (lim(H1,X) (#) lim(H2,X)).x = (lim(H1,X)).x * (lim(H2,X)).x by
VALUED_1:5
      .= lim(H1#x) * (lim(H2,X)).x by A1,A21,Def13
      .= lim(H1#x) * lim(H2#x) by A2,A20,Def13
      .= lim((H1#x) (#) (H2#x)) by A23,SEQ_2:15
      .= lim((H1(#)H2)#x) by A1,A2,A22,Th33;
  end;
A24: now
    let x;
    assume
A25: x in dom (lim(H1,X)-lim(H2,X));
    then
A26: x in (dom lim(H1,X))/\(dom lim(H2,X)) by VALUED_1:12;
    then
A27: x in (dom lim(H2,X)) by XBOOLE_0:def 4;
A28: x in (dom lim(H1,X)) by A26,XBOOLE_0:def 4;
    then
A29: x in X by A1,Def13;
    then
A30: H1#x is convergent & H2#x is convergent by A1,A2,Th19;
    thus (lim(H1,X) - lim(H2,X)).x = (lim(H1,X)).x - (lim(H2,X)).x by A25,
VALUED_1:13
      .= lim(H1#x) - (lim(H2,X)).x by A1,A28,Def13
      .= lim(H1#x) - lim(H2#x) by A2,A27,Def13
      .= lim((H1#x) - (H2#x)) by A30,SEQ_2:12
      .= lim((H1-H2)#x) by A1,A2,A29,Th33;
  end;
  dom (lim(H1,X)+lim(H2,X))=(dom lim(H1,X)) /\ (dom lim(H2,X)) by
VALUED_1:def 1
    .= X /\ (dom lim(H2,X)) by A1,Def13
    .= X /\ X by A2,Def13
    .= X;
  hence lim(H1+H2,X) = lim(H1,X) + lim(H2,X) by A8,A9,Def13;
  X common_on_dom H1-H2 by A7,Th36;
  hence
A31: H1-H2 is_point_conv_on X by A5,Th19;
  dom (lim(H1,X)-lim(H2,X))=(dom lim(H1,X)) /\ (dom lim(H2,X)) by VALUED_1:12
    .= X /\ (dom lim(H2,X)) by A1,Def13
    .= X /\ X by A2,Def13
    .= X;
  hence lim(H1-H2,X) = lim(H1,X) - lim(H2,X) by A31,A24,Def13;
  X common_on_dom H1 (#)H2 by A7,Th36;
  hence
A32: H1(#)H2 is_point_conv_on X by A16,Th19;
  dom (lim(H1,X)(#)lim(H2,X))=(dom lim(H1,X)) /\ (dom lim(H2,X)) by
VALUED_1:def 4
    .= X /\ (dom lim(H2,X)) by A1,Def13
    .= X /\ X by A2,Def13
    .= X;
  hence thesis by A32,A18,Def13;
end;
