
theorem Th7:
  for X be RealNormSpace for f,g,h be Element of
  BoundedLinearOperators(X,X) holds f*(g*h) =(f*g)*h
proof
  let X be RealNormSpace;
  let f,g,h be Element of BoundedLinearOperators(X,X);
  modetrans((g*h),X,X) =modetrans(g,X,X)*modetrans(h,X,X) by LOPBAN_1:def 11;
  then
  modetrans(f,X,X)*modetrans((g*h),X,X) =(modetrans(f,X,X)*modetrans(g,X,X
  ))*modetrans(h,X,X) by Th5;
  hence thesis by LOPBAN_1:def 11;
end;
