let X be RealNormSpace; :: thesis: for f, g being Element of BoundedLinearOperators X,X
for a being Real holds a * (f * g) = (a * f) * g
let f, g be Element of BoundedLinearOperators X,X; :: thesis: for a being Real holds a * (f * g) = (a * f) * g
let a be Real; :: thesis: a * (f * g) = (a * f) * g
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 gg = g as Element of 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)) #) ;
A1: a * (f * g) = (a * 1) * (f * g)
.= (a * f) * (1 * g) by Th11 ;
1 * g = 1 * gg
.= g by RLVECT_1:def 9 ;
hence a * (f * g) = (a * f) * g by A1; :: thesis: verum