reserve S,T,W,Y for RealNormSpace;
reserve f,f1,f2 for PartFunc of S,T;
reserve Z for Subset of S;
reserve i,n for Nat;

theorem
  for X,Y,Z,W be RealNormSpace
  for f be Element of BoundedLinearOperators(Z,W),
      g be Element of BoundedLinearOperators(Y,Z),
      h be Element of BoundedLinearOperators(X,Y)
  holds f*(g*h) = (f*g)*h
  proof
    let X,Y,Z,W be RealNormSpace;
    let f be Element of BoundedLinearOperators(Z,W),
        g be Element of BoundedLinearOperators(Y,Z),
        h be Element of BoundedLinearOperators(X,Y);
    modetrans((g*h),X,Z)
      = modetrans(g,Y,Z) * modetrans(h,X,Y) by LOPBAN_1:def 11; then
    modetrans(f,Z,W) * modetrans((g*h),X,Z)
      = (modetrans(f,Z,W) * modetrans(g,Y,Z)) * modetrans(h,X,Y)
        by RELAT_1:36;
    hence thesis by LOPBAN_1:def 11;
  end;
