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, v3 being Element of V
for f being Function of V,GF holds f (#) <*v1,v2,v3*> = <*((f . v1) * v1),((f . v2) * v2),((f . v3) * v3)*>
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, v3 being Element of V
for f being Function of V,GF holds f (#) <*v1,v2,v3*> = <*((f . v1) * v1),((f . v2) * v2),((f . v3) * v3)*>
let v1, v2, v3 be Element of V; for f being Function of V,GF holds f (#) <*v1,v2,v3*> = <*((f . v1) * v1),((f . v2) * v2),((f . v3) * v3)*>
let f be Function of V,GF; f (#) <*v1,v2,v3*> = <*((f . v1) * v1),((f . v2) * v2),((f . v3) * v3)*>
A1: len (f (#) <*v1,v2,v3*>) =
len <*v1,v2,v3*>
by Def5
.=
3
by FINSEQ_1:45
;
then A2:
dom (f (#) <*v1,v2,v3*>) = {1,2,3}
by FINSEQ_1:def 3, FINSEQ_3:1;
3 in {1,2,3}
by ENUMSET1:def 1;
then A3: (f (#) <*v1,v2,v3*>) . 3 =
(f . (<*v1,v2,v3*> /. 3)) * (<*v1,v2,v3*> /. 3)
by A2, Def5
.=
(f . (<*v1,v2,v3*> /. 3)) * v3
by FINSEQ_4:18
.=
(f . v3) * v3
by FINSEQ_4:18
;
2 in {1,2,3}
by ENUMSET1:def 1;
then A4: (f (#) <*v1,v2,v3*>) . 2 =
(f . (<*v1,v2,v3*> /. 2)) * (<*v1,v2,v3*> /. 2)
by A2, Def5
.=
(f . (<*v1,v2,v3*> /. 2)) * v2
by FINSEQ_4:18
.=
(f . v2) * v2
by FINSEQ_4:18
;
1 in {1,2,3}
by ENUMSET1:def 1;
then (f (#) <*v1,v2,v3*>) . 1 =
(f . (<*v1,v2,v3*> /. 1)) * (<*v1,v2,v3*> /. 1)
by A2, Def5
.=
(f . (<*v1,v2,v3*> /. 1)) * v1
by FINSEQ_4:18
.=
(f . v1) * v1
by FINSEQ_4:18
;
hence
f (#) <*v1,v2,v3*> = <*((f . v1) * v1),((f . v2) * v2),((f . v3) * v3)*>
by A1, A4, A3, FINSEQ_1:45; verum