let V be RealLinearSpace; :: thesis: for v1, v2, v3 being VECTOR of V
for f being Function of the carrier of V,REAL holds f (#) <*v1,v2,v3*> = <*((f . v1) * v1),((f . v2) * v2),((f . v3) * v3)*>

let v1, v2, v3 be VECTOR of V; :: thesis: for f being Function of the carrier of V,REAL holds f (#) <*v1,v2,v3*> = <*((f . v1) * v1),((f . v2) * v2),((f . v3) * v3)*>
let f be Function of the carrier of V,REAL; :: thesis: f (#) <*v1,v2,v3*> = <*((f . v1) * v1),((f . v2) * v2),((f . v3) * v3)*>
A1: len (f (#) <*v1,v2,v3*>) = len <*v1,v2,v3*> by Def7
.= 3 by FINSEQ_1:45 ;
then A2: dom (f (#) <*v1,v2,v3*>) = {1,2,3} by ;
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
.= (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
.= (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
.= (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 ; :: thesis: verum