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 v1, v2 being Element of V
for f being Function of V,GF holds f (#) <*v1,v2*> = <*((f . v1) * v1),((f . v2) * v2)*>
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 v1, v2 being Element of V
for f being Function of V,GF holds f (#) <*v1,v2*> = <*((f . v1) * v1),((f . v2) * v2)*>
let v1, v2 be Element of V; :: thesis: for f being Function of V,GF holds f (#) <*v1,v2*> = <*((f . v1) * v1),((f . v2) * v2)*>
let f be Function of V,GF; :: thesis: f (#) <*v1,v2*> = <*((f . v1) * v1),((f . v2) * v2)*>
A1: len (f (#) <*v1,v2*>) = len <*v1,v2*> by Def5
.= 2 by FINSEQ_1:44 ;
then A2: dom (f (#) <*v1,v2*>) = {1,2} by FINSEQ_1:2, FINSEQ_1:def 3;
2 in {1,2} by TARSKI:def 2;
then A3: (f (#) <*v1,v2*>) . 2 = (f . (<*v1,v2*> /. 2)) * (<*v1,v2*> /. 2) by A2, Def5
.= (f . (<*v1,v2*> /. 2)) * v2 by FINSEQ_4:17
.= (f . v2) * v2 by FINSEQ_4:17 ;
1 in {1,2} by TARSKI:def 2;
then (f (#) <*v1,v2*>) . 1 = (f . (<*v1,v2*> /. 1)) * (<*v1,v2*> /. 1) by A2, Def5
.= (f . (<*v1,v2*> /. 1)) * v1 by FINSEQ_4:17
.= (f . v1) * v1 by FINSEQ_4:17 ;
hence f (#) <*v1,v2*> = <*((f . v1) * v1),((f . v2) * v2)*> by A1, A3, FINSEQ_1:44; :: thesis: verum