let GF be non empty right_complementable well-unital distributive Abelian add-associative right_zeroed associative doubleLoopStr ; :: thesis: for V being non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ModuleStr over GF

for f being Function of V,GF holds f (#) (<*> the carrier of V) = <*> the carrier of V

let V be non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ModuleStr over GF; :: thesis: for f being Function of V,GF holds f (#) (<*> the carrier of V) = <*> the carrier of V

let f be Function of V,GF; :: thesis: f (#) (<*> the carrier of V) = <*> the carrier of V

len (f (#) (<*> the carrier of V)) = len (<*> the carrier of V) by Def5

.= 0 ;

hence f (#) (<*> the carrier of V) = <*> the carrier of V ; :: thesis: verum

for f being Function of V,GF holds f (#) (<*> the carrier of V) = <*> the carrier of V

let V be non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ModuleStr over GF; :: thesis: for f being Function of V,GF holds f (#) (<*> the carrier of V) = <*> the carrier of V

let f be Function of V,GF; :: thesis: f (#) (<*> the carrier of V) = <*> the carrier of V

len (f (#) (<*> the carrier of V)) = len (<*> the carrier of V) by Def5

.= 0 ;

hence f (#) (<*> the carrier of V) = <*> the carrier of V ; :: thesis: verum