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 E,F,G be RealNormSpace,
          f be Point of R_NormSpace_of_BoundedLinearOperators (E,F),
          g be Point of R_NormSpace_of_BoundedLinearOperators (F,G)
  holds
    ex h be Point of R_NormSpace_of_BoundedLinearOperators (E,G)
    st h = g*f
     & ||.h.|| <= ||.g.||*||.f.||
  proof
    let E,F,G be RealNormSpace,
            f be Point of R_NormSpace_of_BoundedLinearOperators (E,F),
            g be Point of R_NormSpace_of_BoundedLinearOperators (F,G);
    reconsider Lf=f as Lipschitzian LinearOperator of E,F by LOPBAN_1:def 9;
    reconsider Lg=g as Lipschitzian LinearOperator of F,G by LOPBAN_1:def 9;
    set Lh = Lg*Lf;
    reconsider Lh as Lipschitzian LinearOperator of E,G by LOPBAN_2:2;
    reconsider h = Lh as Point of R_NormSpace_of_BoundedLinearOperators (E,G)
        by LOPBAN_1:def 9;
    take h;
    thus h = g * f;
    A8: ||.h.|| = upper_bound (PreNorms(modetrans(h,E,G))) by LOPBAN_1:def 13
               .= upper_bound PreNorms(Lh) by LOPBAN_1:29;
    for t be Real st t in PreNorms(Lh)
    holds t <= ||.g.|| * ||.f.||
    proof
      let t be Real;
      assume t in PreNorms(Lh); then
      consider w be Point of E such that
      A9: t = ||. Lh.w .|| & ||.w.|| <= 1;
      A10: ||. Lh.w .|| = ||.Lg.(Lf.w).|| by FUNCT_2:15;
      A11: ||. Lf.w .|| <= ||.f.|| * ||.w.|| by LOPBAN_1:32;
      ||.f.|| * ||.w.|| <= ||.f.|| * 1 by A9,XREAL_1:64; then
      A12: ||. Lf.w .|| <= ||.f.|| by A11,XXREAL_0:2;
      A13: ||.Lg. (Lf.w) .|| <= ||.g.|| * ||. (Lf.w) .|| by LOPBAN_1:32;
      ||.g.|| * ||. (Lf.w) .|| <= ||.g.|| * ||.f.|| by A12,XREAL_1:64;
      hence thesis by A9,A10,A13,XXREAL_0:2;
    end;
    hence ||.h.|| <= ||.g.|| * ||.f.|| by A8,SEQ_4:45;
  end;
