
theorem Th12:
  for X be ComplexNormSpace for f,g be Element of
  BoundedLinearOperators(X,X) for a be Complex holds a*(f*g) =(a*f)*g
proof
  let X be ComplexNormSpace;
  let f,g be Element of BoundedLinearOperators(X,X);
  let a be Complex;
  set RRL=CLSStruct (# BoundedLinearOperators(X,X), Zero_(
    BoundedLinearOperators(X,X), C_VectorSpace_of_LinearOperators(X,X)), Add_(
    BoundedLinearOperators(X,X), C_VectorSpace_of_LinearOperators(X,X)), Mult_(
    BoundedLinearOperators(X,X), C_VectorSpace_of_LinearOperators(X,X)) #);
  reconsider gg=g as Element of RRL;
A1: (1r*g)=1r*gg .=g by CLVECT_1:def 5;
  a*(f*g)=(a*1r)*(f*g) by COMPLEX1:def 4
    .=(a*f)*(1r*g) by Th11;
  hence thesis by A1;
end;
