reserve x,y for set,
  i for Nat;
reserve V for non empty CLSStruct,
  u,v,v1,v2,v3 for VECTOR of V,
  A for Subset of V,
  l, l1, l2 for C_Linear_Combination of A,
  x,y,y1,y2 for set,
  a,b for Complex,
  F for FinSequence of the carrier of V,
  f for Function of the carrier of V, COMPLEX;

theorem Th10:
  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 FINSEQ_3:1;
  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 FINSEQ_3:1;
  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 FINSEQ_3:1;
  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;
