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 Th28:
  {x} common_on_dom H implies (||.H.||)#x = ||.H#x .|| & (-H)#x = (-1)*(H#x)
  proof
    assume
    AS1: {x} common_on_dom H;
    now
      let n be Element of NAT;
      x in {x} & {x} c= dom(H.n) by AS1, TARSKI:def 1;
      then x in dom (H.n);
      then
      A2: x in dom ||.(H.n).|| by NORMSP_0:def 2;
      thus ((||.H.||)#x).n = (||.H.||.n).x by SEQFUNC:def 10
      .= ||.(H.n).||.x by Def4
      .= ||.(H.n)/.x.|| by A2, NORMSP_0:def 2  
      .= ||.(H#x).n.|| by Def10
      .= ||.(H#x).|| .n by NORMSP_0:def 4;
    end;
    hence (||.H.||)#x = ||. H#x .|| by FUNCT_2:63;

    now
      let n be Element of NAT;
      x in {x} & {x} c= dom(H.n) by AS1, TARSKI:def 1;
      then x in dom (H.n);
      then
      A2: x in dom (-(H.n)) by VFUNCT_1:def 5;
      thus ((-H)#x).n = ((-H).n)/.x by Def10
      .= (-H.n)/.x by Def3
      .= -((H.n)/.x) by A2, VFUNCT_1:def 5
      .= -((H#x).n) by Def10
      .=(-1)*((H#x).n) by RLVECT_1:16
      .= ((-1)*(H#x)).n by NORMSP_1:def 5;
    end;
    hence thesis by FUNCT_2:63;
  end;
