
theorem FINSEQ940B:
  for a1,a2,a3,b1,b2,b3 be Complex, n be Nat,
      f,g be n-element complex-valued FinSequence holds
  (f^<*a1,a2,a3*>) + (g^<*b1,b2,b3*>) = (f+g)^<*a1+b1,a2+b2,a3+b3*>
  proof
    let a1,a2,a3,b1,b2,b3 be Complex, n be Nat,
        f,g be n-element complex-valued FinSequence;
    <*a1,a2,a3*>=<*a1*>^(<*a2*>^<*a3*>) &
    <*b1,b2,b3*>=<*b1*>^(<*b2*>^<*b3*>) by FINSEQ_1:32;then
    A2: f^<*a1,a2,a3*> = f^<*a1*>^(<*a2*>^<*a3*>) &
    g^<*b1,b2,b3*> = g^<*b1*>^(<*b2*>^<*b3*>) by FINSEQ_1:32;
    (f+g)^<*a1+b1,a2+b2,a3+b3*> = (f+g)^(<*a1+b1*>^(<*a2+b2*>^<*a3+b3*>))
      by FINSEQ_1:32
    .= ((f+g)^<*a1+b1*>)^(<*a2+b2*>^<*a3+b3*>) by FINSEQ_1:32
    .= ((f^<*a1*>)+(g^<*b1*>))^(<*a2+b2*>^<*a3+b3*>) by FINSEQ_9:40
    .= ((f^<*a1*>)+(g^<*b1*>))^<*a2+b2*>^<*a3+b3*> by FINSEQ_1:32
    .= (((f^<*a1*>)^<*a2*>)+((g^<*b1*>)^<*b2*>))^<*a3+b3*> by FINSEQ_9:40
    .= ((f^<*a1*>)^<*a2*>^<*a3*>)+((g^<*b1*>)^<*b2*>^<*b3*>) by FINSEQ_9:40
    .= ((f^<*a1*>)^<*a2*>^<*a3*>)+((g^<*b1*>)^(<*b2*>^<*b3*>)) by FINSEQ_1:32
    .= ((f^<*a1*>)^(<*a2*>^<*a3*>))+((g^<*b1*>)^(<*b2*>^<*b3*>))
      by FINSEQ_1:32;
    hence thesis by A2;
  end;
