
theorem FINSEQ940A:
  for a1,a2,b1,b2 be Complex, n be Nat,
      f,g be n-element complex-valued FinSequence holds
  (f^<*a1,a2*>) + (g^<*b1,b2*>) = (f+g)^<*a1+b1,a2+b2*>
  proof
    let a1,a2,b1,b2 be Complex, n be Nat,
        f,g be n-element complex-valued FinSequence;
    A1: f^<*a1,a2*> = f^<*a1*>^<*a2*> &
    g^<*b1,b2*> = g^<*b1*>^<*b2*> by FINSEQ_1:32;
    (f+g)^<*a1+b1,a2+b2*> = ((f+g)^<*a1+b1*>)^<*a2+b2*> by FINSEQ_1:32
    .= ((f^<*a1*>)+(g^<*b1*>))^<*a2+b2*> by FINSEQ_9:40
    .= ((f^<*a1*>)^<*a2*>)+((g^<*b1*>)^<*b2*>) by FINSEQ_9:40;
    hence thesis by A1;
  end;
