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 be RealNormSpace
  for f be Element of R_NormSpace_of_BoundedLinearOperators(Y,Z),
      g be Element of R_NormSpace_of_BoundedLinearOperators(X,Y)
  for a be Real
  holds a*(f*g) = (a*f)*g
  proof
    let X,Y,Z be RealNormSpace;
    let f be Element of R_NormSpace_of_BoundedLinearOperators(Y,Z),
        g be Element of R_NormSpace_of_BoundedLinearOperators(X,Y);
    let a be Real;
    reconsider jj = 1 as Real;
    a*(f*g) = (a*jj) * (f*g)
           .= (a*f) * (jj*g) by LPB2Th11;
    hence thesis by RLVECT_1:def 8;
  end;
