reserve X,Y,Z for non trivial RealBanachSpace;

theorem XXXX:
  for X,Y be RealNormSpace,
      v be Lipschitzian LinearOperator of X,Y,
      w be Point of R_NormSpace_of_BoundedLinearOperators(X,Y),
      a be Real
  st v = w
  holds a*w = a(#)v
  proof
    let X,Y be RealNormSpace,
        v be Lipschitzian LinearOperator of X,Y,
        w be Point of R_NormSpace_of_BoundedLinearOperators(X,Y),
        a be Real;
    set S = R_Normed_Algebra_of_BoundedLinearOperators X;
    assume
    A1: v = w;
    a*w is Lipschitzian LinearOperator of X,Y by LOPBAN_1:def 9; then
    A3: dom (a*w) = the carrier of X by FUNCT_2:def 1;
    A4: dom (a(#)v) = dom v by VFUNCT_1:def 4
     .= the carrier of X by FUNCT_2:def 1;
    for s being object st s in dom (a(#)v)
    holds (a(#)v) . s = (a*w) . s
    proof
      let s be object;
      assume
      A5: s in dom (a(#)v); then
      reconsider d = s as Point of X;
      thus (a(#)v) . s = (a(#)v) /.d by A4,PARTFUN1:def 6
      .= a*v/.d by A5,VFUNCT_1:def 4
      .= (a*w).s by A1,LOPBAN_1:36;
    end;
    hence a(#)v = a*w by A3,A4,FUNCT_1:2;
  end;
