reserve x,y,y1,y2 for set,
  p for FinSequence,
  i,k,l,n for Nat,
  V for RealLinearSpace,
  u,v,v1,v2,v3,w for VECTOR of V,
  a,b for Real,
  F,G,H1,H2 for FinSequence of V,
  A,B for Subset of V,
  f for Function of the carrier of V, REAL;
reserve K,L,L1,L2,L3 for Linear_Combination of V;
reserve l,l1,l2 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 Def7
    .= 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
,Def7
    .= 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
,Def7
    .= 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
,Def7
    .= 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;
