reserve X,Y,Z for non trivial RealBanachSpace;

theorem RELAT136:
  for X,Y,Z,W be RealNormSpace,
      f be Point of R_NormSpace_of_BoundedLinearOperators(X,Y),
      g be Point of R_NormSpace_of_BoundedLinearOperators(Y,Z),
      h be Point of R_NormSpace_of_BoundedLinearOperators(Z,W)
  holds h*(g*f) = (h*g)*f
  proof
    let X,Y,Z,W be RealNormSpace,
        f be Point of R_NormSpace_of_BoundedLinearOperators(X,Y),
        g be Point of R_NormSpace_of_BoundedLinearOperators(Y,Z),
        h be Point of R_NormSpace_of_BoundedLinearOperators(Z,W);
    A2: h*g is Lipschitzian LinearOperator of Y,W by LOPBAN_2:2;
    g*f is Lipschitzian LinearOperator of X,Z by LOPBAN_2:2;
    hence h*(g*f)
       = modetrans(h,Z,W) * (modetrans(g,Y,Z) * modetrans(f,X,Y))
          by LOPBAN_1:29
      .= (modetrans(h,Z,W) * modetrans(g,Y,Z)) * modetrans(f,X,Y) by RELAT_1:36
      .= (h*g)*f by A2,LOPBAN_1:29;
  end;
