let GF be non empty right_complementable well-unital distributive Abelian add-associative right_zeroed associative doubleLoopStr ; 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; 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; for f being Function of V,GF holds f (#) <*v1,v2*> = <*((f . v1) * v1),((f . v2) * v2)*>
let f be Function of V,GF; 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; verum