theorem
  f (#) <* v1,v2,v3 *> = <* v1 * f.v1, v2 * f.v2, v3 * f.v3 *>
proof
A1: len(f (#) <* v1,v2,v3 *>) = len<* v1,v2,v3 *> by Def6
    .= 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 = (<* v1,v2,v3 *>/.3) * f.(<* v1,v2,v3 *>/.3)
  by A2,Def6
    .= v3 * f.(<* v1,v2,v3 *>/.3) by FINSEQ_4:18
    .= v3 * f.v3 by FINSEQ_4:18;
  2 in {1,2,3} by ENUMSET1:def 1;
  then
A4: (f (#) <* v1,v2,v3 *>).2 = (<* v1,v2,v3 *>/.2) * f.(<* v1,v2,v3 *>/.2)
  by A2,Def6
    .= v2 * f.(<* v1,v2,v3 *>/.2) by FINSEQ_4:18
    .= v2 * f.v2 by FINSEQ_4:18;
  1 in {1,2,3} by ENUMSET1:def 1;
  then (f (#) <* v1,v2,v3 *>).1 = (<* v1,v2,v3 *>/.1) * f.(<* v1,v2,v3 *>/.1)
  by A2,Def6
    .= v1 * f.(<* v1,v2,v3 *>/.1) by FINSEQ_4:18
    .= v1 * f.v1 by FINSEQ_4:18;
  hence thesis by A1,A4,A3,FINSEQ_1:45;
end;
