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 Th8:
  f (#) <* v *> = <* f.v * v *>
proof
A1: 1 in {1} by FINSEQ_1:2;
A2: len(f (#) <* v *>) = len<* v *> by Def5
    .= 1 by FINSEQ_1:40;
  then dom(f (#)<* v *>) = {1} by FINSEQ_1:2,def 3;
  then (f (#) <* v *>).1 = f.(<* v *>/.1) * <* v *>/.1 by A1,Def5
    .= f.(<* v *>/.1) * v by FINSEQ_4:16
    .= f.v * v by FINSEQ_4:16;
  hence thesis by A2,FINSEQ_1:40;
end;
