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

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) 
  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 NORMSP_1:19;
      hence (H1+H2)#x is convergent by A1, A2, A4, Th30;
    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 NORMSP_1:20;
      hence (H1-H2)#x is convergent by A1, A2, A6, Th30;
    end;

    thus
    A8: H1+H2 is_point_conv_on X by A1, A2, A3, Th36, 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 VFUNCT_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;
      X15: (lim(H1, X)).x =(lim(H1, X))/.x by A13, PARTFUN1:def 6;
      X16: (lim(H2, X)).x =(lim(H2, X))/.x by A12, PARTFUN1:def 6;
      thus (lim(H1, X) + lim(H2, X)).x = (lim(H1, X) + lim(H2, X))/.x
      by A10, PARTFUN1:def 6
      .= (lim(H1, X))/.x + (lim(H2, X))/.x by A10, VFUNCT_1:def 1
      .= lim(H1#x) + (lim(H2, X))/.x by X15, A1, A13, Def13
      .= lim(H1#x) + lim(H2#x) by X16, A2, A12, Def13
      .= lim((H1#x) + (H2#x)) by A15, NORMSP_1:25
      .= lim((H1+H2)#x) by A1, A2, A14, Th30;
    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 VFUNCT_1:def 2;
      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;
      X15: (lim(H1, X)).x =(lim(H1, X))/.x by A28, PARTFUN1:def 6;
      X16: (lim(H2, X)).x =(lim(H2, X))/.x by A27, PARTFUN1:def 6;
      thus (lim(H1, X) - lim(H2, X)).x
      = (lim(H1, X) - lim(H2, X))/.x by A25, PARTFUN1:def 6
      .= (lim(H1, X))/.x - (lim(H2, X))/.x by A25, VFUNCT_1:def 2
      .= lim(H1#x) - (lim(H2, X))/.x by X15, A1, A28, Def13
      .= lim(H1#x) - lim(H2#x) by X16, A2, A27, Def13
      .= lim((H1#x) - (H2#x)) by A30, NORMSP_1:26
      .= lim((H1-H2)#x) by A1, A2, A29, Th30;
    end;

    dom (lim(H1, X)+lim(H2, X))=(dom lim(H1, X)) /\ (dom lim(H2, X))
    by VFUNCT_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 A1, A2, 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 VFUNCT_1:def 2
    .= 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;
  end;
