
theorem Th12:
  for X be RealNormSpace for f,g be Element of
  BoundedLinearOperators(X,X) for a be Real holds a*(f*g) =(a*f)*g
proof
  let X be RealNormSpace;
  let f,g be Element of BoundedLinearOperators(X,X);
  let a be Real;
  set RRL=RLSStruct (# BoundedLinearOperators(X,X), Zero_(
    BoundedLinearOperators(X,X), R_VectorSpace_of_LinearOperators(X,X)), Add_(
    BoundedLinearOperators(X,X), R_VectorSpace_of_LinearOperators(X,X)), Mult_(
    BoundedLinearOperators(X,X), R_VectorSpace_of_LinearOperators(X,X)) #);
  reconsider jj=1 as Real;
  reconsider gg=g as Element of RRL;
A1: (jj*g)=jj*gg .=g by RLVECT_1:def 8;
  a*(f*g)=(a*jj)*(f*g) .=(a*f)*(jj*g) by Th11;
  hence thesis by A1;
end;
