reserve X,Y,Z for non trivial RealBanachSpace;

theorem NRM:
  for X,Y,Z be RealNormSpace,
      u be Point of R_NormSpace_of_BoundedLinearOperators(X,Y),
      v be Point of R_NormSpace_of_BoundedLinearOperators(Y,Z),
      w be Point of R_NormSpace_of_BoundedLinearOperators(X,Z)
   st w = v*u
  holds ||.w.|| <= ||.v.|| * ||.u.||
  proof
    let X,Y,Z be RealNormSpace,
        u be Point of R_NormSpace_of_BoundedLinearOperators(X,Y),
        v be Point of R_NormSpace_of_BoundedLinearOperators(Y,Z),
        w be Point of R_NormSpace_of_BoundedLinearOperators(X,Z);
    assume
    A2: w = v*u;
    modetrans(v,Y,Z) = v & modetrans(u,X,Y) = u by LOPBAN_1:def 11;
    hence ||.w.|| <= ||.v.|| * ||.u.|| by A2,LOPBAN_2:2;
  end;
