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 Th29:
  {x} common_on_dom H implies (r(#)H)#x = r*(H#x)
  proof
    assume
    A1: {x} common_on_dom H;

    now
      let n be Element of NAT;

      x in {x} & {x} c= dom(H.n) by A1, TARSKI:def 1;
      then x in dom (H.n);
      then
      A2: x in dom (r(#)(H.n)) by VFUNCT_1:def 4;
      X2: (r(#)H).n = r(#)(H.n) by Def1;
      thus ((r(#)H)#x).n = ((r(#)H).n)/.x by Def10
      .= r*((H.n)/.x) by X2, A2, VFUNCT_1:def 4
      .= r*((H#x).n) by Def10
      .= (r*(H#x)).n by NORMSP_1:def 5;
    end;
    hence thesis by FUNCT_2:63;
  end;
