
theorem APB:
  for a,b be Complex holds <*a*>+<*b*> = <*(a+b)*>
  proof
    let a,b be Complex;
    dom <*a*> = Seg 1 & dom <*b*> = Seg 1 & dom <*(a+b)*> = Seg 1
      by FINSEQ_1:def 8; then
    A2: dom (<*a*>+<*b*>) = (Seg 1)/\(Seg 1) by VALUED_1:def 1
    .= Seg 1; then
    1 in dom (<*a*>+<*b*>); then
    (<*a*>+<*b*>).1 = <*a*>.1+<*b*>.1 by VALUED_1:def 1;
    hence thesis by A2,FINSEQ_1:def 8;
  end;
