let X be ComplexNormSpace; :: thesis: for f, g being Element of BoundedLinearOperators (X,X)
for a being Complex holds a * (f * g) = (a * f) * g
let f, g be Element of BoundedLinearOperators (X,X); :: thesis: for a being Complex holds a * (f * g) = (a * f) * g
let a be Complex; :: thesis: a * (f * g) = (a * f) * g
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 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)))) #) ;
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 a * (f * g) = (a * f) * g by A1; :: thesis: verum