
theorem A7B7:
  for a1,a2,a3,a4,a5,a6,a7,b1,b2,b3,b4,b5,b6,b7 be Complex holds
    <*a1,a2,a3,a4,a5,a6,a7*> + <*b1,b2,b3,b4,b5,b6,b7*> =
      <*a1+b1,a2+b2,a3+b3,a4+b4,a5+b5,a6+b6,a7+b7*>
  proof
    let a1,a2,a3,a4,a5,a6,a7,b1,b2,b3,b4,b5,b6,b7 be Complex;
    A1: <*a1,a2,a3,a4,a5*> + <*b1,b2,b3,b4,b5*> =
      <*a1+b1,a2+b2,a3+b3,a4+b4,a5+b5*> by A5B5;
    <*a1,a2,a3,a4,a5,a6,a7*> = <*a1,a2,a3,a4,a5*>^<*a6,a7*> &
    <*b1,b2,b3,b4,b5,b6,b7*> = <*b1,b2,b3,b4,b5*>^<*b6,b7*>
      by AOFA_A00:def 5; then
    <*a1,a2,a3,a4,a5,a6,a7*> + <*b1,b2,b3,b4,b5,b6,b7*> =
      <*a1+b1,a2+b2,a3+b3,a4+b4,a5+b5*>^<*a6+b6,a7+b7*> by A1,FINSEQ940A;
    hence thesis by AOFA_A00:def 5;
  end;
