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;

theorem Th27:
  {x} common_on_dom H1 & {x} common_on_dom H2
  implies H1#x + H2#x = (H1+H2)#x & H1#x - H2#x = (H1-H2)#x 
  proof
    assume that
    A1: {x} common_on_dom H1 and
    A2: {x} common_on_dom H2;

    now
      let n be Element of NAT;
      A3: {x} c= dom (H2.n) by A2;
      x in {x} & {x} c= dom(H1.n) by A1, TARSKI:def 1;
      then x in (dom(H1.n) /\ dom(H2.n)) by A3, XBOOLE_0:def 4;
      then
      A4: x in dom(H1.n + H2.n) by VFUNCT_1:def 1;
      X4: dom(H1.n + H2.n) = dom ((H1+H2).n) by Def5;
      thus (H1#x + H2#x).n = (H1#x).n + (H2#x).n by NORMSP_1:def 2
      .= (H1.n)/.x + (H2#x).n by Def10
      .= (H1.n)/.x + (H2.n)/.x by Def10
      .= ((H1.n) + (H2.n))/.x by A4, VFUNCT_1:def 1
      .= ((H1.n) + (H2.n)).x by A4, PARTFUN1:def 6
      .= ((H1+H2).n).x by Def5
      .= ((H1+H2).n)/.x by A4, X4, PARTFUN1:def 6
      .= ((H1+H2)#x).n by Def10;
    end;
    hence H1#x + H2#x = (H1+H2)#x by FUNCT_2:63;
    now
      let n be Element of NAT;
      A5: {x} c= dom (H2.n) by A2;
      x in {x} & {x} c= dom(H1.n) by A1, TARSKI:def 1;
      then x in (dom(H1.n) /\ dom(H2.n)) by A5, XBOOLE_0:def 4;
      then
      A6: x in dom(H1.n - H2.n) by VFUNCT_1:def 2;
      X6: dom (H1.n - H2.n) = dom ((H1-H2).n) by Th3;
      thus (H1#x - H2#x).n = (H1#x).n - (H2#x).n by NORMSP_1:def 3
      .= (H1.n)/.x - (H2#x).n by Def10
      .= (H1.n)/.x - (H2.n)/.x by Def10
      .= ((H1.n) - (H2.n))/.x by A6, VFUNCT_1:def 2
      .= ((H1.n) - (H2.n)).x by A6, PARTFUN1:def 6
      .= ((H1-H2).n).x by Th3
      .= ((H1-H2).n)/.x by X6, A6, PARTFUN1:def 6
      .= ((H1-H2)#x).n by Def10;
    end;
    hence H1#x - H2#x = (H1-H2)#x by FUNCT_2:63;
  end;
