
theorem
  for a,b be Complex, n be non trivial Nat,
      f be n-element complex-valued FinSequence holds
        <*a,b*>(#)f = <*a*f.1,b*f.2*>
  proof
    let a,b be Complex, n be non trivial Nat,
        f be n-element complex-valued FinSequence;
    reconsider g = <*a,b*> as 2-element complex-valued FinSequence;
    A1: len (g(#)f) = 2 by CARD_1:def 7; then
    1 in dom (g(#)f) & 2 in dom (g(#)f) by FINSEQ_3:25; then
    (g(#)f).1 = g.1*f.1 & (g(#)f).2 = g.2*f.2 by VALUED_1:def 4;
    hence thesis by A1,FINSEQ_1:44;
  end;
