let R be Ring; :: thesis: for V being RightMod of R
for v1, v2 being Vector of V
for f being Function of V,R holds f (#) <*v1,v2*> = <*(v1 * (f . v1)),(v2 * (f . v2))*>
let V be RightMod of R; :: thesis: for v1, v2 being Vector of V
for f being Function of V,R holds f (#) <*v1,v2*> = <*(v1 * (f . v1)),(v2 * (f . v2))*>
let v1, v2 be Vector of V; :: thesis: for f being Function of V,R holds f (#) <*v1,v2*> = <*(v1 * (f . v1)),(v2 * (f . v2))*>
let f be Function of V,R; :: thesis: f (#) <*v1,v2*> = <*(v1 * (f . v1)),(v2 * (f . v2))*>
A1: len (f (#) <*v1,v2*>) = len <*v1,v2*> by Def6
.= 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 = (<*v1,v2*> /. 2) * (f . (<*v1,v2*> /. 2)) by A2, Def6
.= v2 * (f . (<*v1,v2*> /. 2)) by FINSEQ_4:17
.= v2 * (f . v2) by FINSEQ_4:17 ;
1 in {1,2} by TARSKI:def 2;
then (f (#) <*v1,v2*>) . 1 = (<*v1,v2*> /. 1) * (f . (<*v1,v2*> /. 1)) by A2, Def6
.= v1 * (f . (<*v1,v2*> /. 1)) by FINSEQ_4:17
.= v1 * (f . v1) by FINSEQ_4:17 ;
hence f (#) <*v1,v2*> = <*(v1 * (f . v1)),(v2 * (f . v2))*> by A1, A3, FINSEQ_1:44; :: thesis: verum