let V be non empty CLSStruct ; :: thesis: for v being VECTOR of V
for f being Function of the carrier of V,COMPLEX holds f (#) <*v*> = <*((f . v) * v)*>
let v be VECTOR of V; :: thesis: for f being Function of the carrier of V,COMPLEX holds f (#) <*v*> = <*((f . v) * v)*>
let f be Function of the carrier of V,COMPLEX; :: thesis: f (#) <*v*> = <*((f . v) * v)*>
A1: 1 in {1} by FINSEQ_1:2;
A2: len (f (#) <*v*>) = len <*v*> by Def5
.= 1 by FINSEQ_1:40 ;
then dom (f (#) <*v*>) = {1} by FINSEQ_1:2, FINSEQ_1:def 3;
then (f (#) <*v*>) . 1 = (f . (<*v*> /. 1)) * (<*v*> /. 1) by A1, Def5
.= (f . (<*v*> /. 1)) * v by FINSEQ_4:16
.= (f . v) * v by FINSEQ_4:16 ;
hence f (#) <*v*> = <*((f . v) * v)*> by A2, FINSEQ_1:40; :: thesis: verum