
theorem A5B5:
  for a1,a2,a3,a4,a5,b1,b2,b3,b4,b5 be Complex holds
  <*a1,a2,a3,a4,a5*> + <*b1,b2,b3,b4,b5*> = <*a1+b1,a2+b2,a3+b3,a4+b4,a5+b5*>
  proof
    let a1,a2,a3,a4,a5,b1,b2,b3,b4,b5 be Complex;
    A1: <*a1,a2,a3,a4*> + <*b1,b2,b3,b4*> = <*a1+b1,a2+b2,a3+b3,a4+b4*>
      by FINSEQ_9:43;
    <*a1,a2,a3,a4,a5*> = <*a1,a2,a3,a4*>^<*a5*> &
    <*b1,b2,b3,b4,b5*> = <*b1,b2,b3,b4*>^<*b5*> by FINSEQ_4:75; then
    <*a1,a2,a3,a4,a5*> + <*b1,b2,b3,b4,b5*> =
      <*a1+b1,a2+b2,a3+b3,a4+b4*>^<*a5+b5*> by A1,FINSEQ_9:40
    .= <*a1+b1,a2+b2,a3+b3,a4+b4,a5+b5*> by FINSEQ_4:75;
    hence thesis;
  end;
