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
  H is_point_conv_on X
  implies ||.H.|| is_point_conv_on X & lim (||.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
      X0: x in X; then
      X1: {x} common_on_dom H by A1, Th25;
      ||. H#x .|| is convergent by X0, A1, Th19, NORMSP_1:23;
      hence ||.H.||#x is convergent by Th28, X1;
    end;

    A3:
    now
      let x;
      assume A30:x in dom (-lim(H, X));
      then
      A4: x in dom lim(H, X) by VFUNCT_1:def 5;
      then
      A5: x in X by A1, Def13;
      X5: lim(H, X) /.x =lim(H, X) .x by A4, PARTFUN1:def 6; 
      X1: {x} common_on_dom H by A5, A1, Th25;
      thus (-lim(H, X)).x = (-lim(H, X))/.x by A30, PARTFUN1:def 6
      .= -(lim(H, X)/.x) by A30, VFUNCT_1:def 5
      .= -(lim(H#x)) by X5, A1, A4, Def13
      .=(-1)* (lim(H#x)) by RLVECT_1:16
      .= lim((-1)*(H#x)) by A1, A5, Th19, NORMSP_1:28
      .= lim((-H)#x) by Th28, X1;
    end;

    thus
    A7: ||.H.|| is_point_conv_on X by A1, Th37, A2, SEQFUNC:20;

    A8:
    now
      let x;
      assume A91: x in dom ||.lim(H, X).||;
      then
      A9: x in dom lim(H, X) by NORMSP_0:def 3;
      then
      A90: x in X by A1, Def13;
      then
      A10: H#x is convergent by A1, Th19;
      X1: {x} common_on_dom H by A90, A1, Th25;
      X10: lim(H, X)/.x = lim(H, X).x by A9, PARTFUN1:def 6;
      thus ||.lim(H, X).||.x = ||.lim(H, X)/.x.|| by A91, NORMSP_0:def 3
      .= ||.lim(H#x).|| by A1, A9, X10, Def13
      .= lim ||.(H#x).|| by A10, LOPBAN_1:20
      .= lim(||.H.||#x) by Th28, X1;
    end;

    A11:
    now
      let x;
      assume
      A90: x in X;
      then X10: (-1)*(H#x) is convergent by A1, Th19, NORMSP_1:22;
      {x} common_on_dom H by A90, A1, Th25;
      hence (-H)#x is convergent by Th28, X10;
    end;

    dom ||.lim(H, X).|| = dom lim(H, X) by NORMSP_0:def 3
    .= X by A1, Def13;
    hence lim (||.H.||, X) = ||. lim (H, X) .|| by A7, A8, SEQFUNC:def 13;
    thus
    A12: -H is_point_conv_on X by A1, Th37, A11, Th19;
    dom (-lim(H, X)) = dom lim(H, X) by VFUNCT_1:def 5
    .= X by A1, Def13;
    hence thesis by A12, A3, Def13;
  end;
