reserve p,q,r for FinSequence,
  x,y,y1,y2 for set,
  i,k for Element of NAT,
  GF for add-associative right_zeroed right_complementable Abelian associative
  well-unital distributive non empty doubleLoopStr,
  V for Abelian
  add-associative right_zeroed right_complementable vector-distributive
  scalar-distributive scalar-associative scalar-unital
   non empty ModuleStr over GF,
  u,v,v1,v2,v3,w for Element of V,
  a,b for Element of GF,
  F,G ,H for FinSequence of V,
  A,B for Subset of V,
  f for Function of V, GF;
reserve L,L1,L2,L3 for Linear_Combination of V;
reserve l for Linear_Combination of A;

theorem
  f (#) <* v1,v2,v3 *> = <* f.v1 * v1, f.v2 * v2, f.v3 * v3 *>
proof
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 thesis by A1,A4,A3,FINSEQ_1:45;
end;
