let GF be non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr ; :: thesis: for V being non empty right_complementable VectSp-like Abelian add-associative right_zeroed VectSpStr of GF
for f being Function of the carrier of V,the carrier of GF holds f (#) (<*> the carrier of V) = <*> the carrier of V
let V be non empty right_complementable VectSp-like Abelian add-associative right_zeroed VectSpStr of GF; :: thesis: for f being Function of the carrier of V,the carrier of GF holds f (#) (<*> the carrier of V) = <*> the carrier of V
let f be Function of the carrier of V,the carrier of GF; :: thesis: f (#) (<*> the carrier of V) = <*> the carrier of V
len (f (#) (<*> the carrier of V)) = len (<*> the carrier of V) by Def8
.= 0 by FINSEQ_1:32 ;
hence f (#) (<*> the carrier of V) = <*> the carrier of V by FINSEQ_1:32; :: thesis: verum