reserve X,Y,Z for non trivial RealBanachSpace;

theorem LOPBAN1624:
  for X,Y,Z be RealNormSpace,
      u be Point of R_NormSpace_of_BoundedLinearOperators(X,Y),
      w be Point of R_NormSpace_of_BoundedLinearOperators(Y,Z),
      r be Real
  holds w*(r*u) = r*w*u & (r*w)*u = r*w*u & r*w*u = r*(w*u)
  proof
    let X,Y,Z be RealNormSpace,
        u be Point of R_NormSpace_of_BoundedLinearOperators(X,Y),
        w be Point of R_NormSpace_of_BoundedLinearOperators(Y,Z),
        r be Real;
    A3: for x be Point of X holds (w*(r*u)).x = r*(w*u).x
    proof
      let x be Point of X;
      thus (w*(r*u)).x
       = modetrans(w,Y,Z).(modetrans(r*u,X,Y).x) by FUNCT_2:15
      .= modetrans(w,Y,Z).((r*u).x ) by LOPBAN_1:def 11
      .= modetrans(w,Y,Z).(r*(u.x)) by LOPBAN_1:36
      .= r * modetrans(w,Y,Z).(u.x) by LOPBAN_1:def 5
      .= r * modetrans(w,Y,Z).(modetrans(u,X,Y).x) by LOPBAN_1:def 11
      .= r * (w*u).x by FUNCT_2:15;
    end;
    for x be Point of X holds ((r*w)*u).x = r*(w*u).x
    proof
      let x be Point of X;
      thus ((r*w)*u).x = modetrans(r*w,Y,Z).(modetrans(u,X,Y).x) by FUNCT_2:15
      .= (r*w).(modetrans(u,X,Y).x ) by LOPBAN_1:def 11
      .= r * w.(modetrans(u,X,Y).x) by LOPBAN_1:36
      .= r * modetrans(w,Y,Z).(modetrans(u,X,Y).x) by LOPBAN_1:def 11
      .= r * (w*u).x by FUNCT_2:15;
    end;
    then r*w*u = r*(w*u) by LOPBAN_1:36;
    hence thesis by A3,LOPBAN_1:36;
  end;
