
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 1;
    hence thesis by A1,FINSEQ_1:44;
  end;
