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: 1 * g = 1 * gg
.= g by RLVECT_1:def 11 ;
a * (f * g) = (a * 1) * (f * g)
.= (a * f) * (1 * g) by Th11 ;
hence a * (f * g) = (a * f) * g by A1; :: thesis: verum