
theorem
  for a,b,c,d,x,y,z,v be Complex holds
    <*a,b,c,d*>+<*x,y,z,v*> = <*a+x,b+y,c+z,d+v*>
  proof
    let a,b,c,d,x,y,z,v be Complex;
    reconsider f = <*a,b,c*> as 3-element complex-valued FinSequence;
    reconsider g = <*x,y,z*> as 3-element complex-valued FinSequence;
    <*a,b,c,d*> = f^<*d*> & <*x,y,z,v*> = g^<*v*> by FINSEQ_4:def 7; then
    <*a,b,c,d*>+<*x,y,z,v*> = (f+g)^<*(d+v)*> by FPA
    .= <*(a+x),(b+y),(c+z)*>^<*(d+v)*> by AP3;
    hence thesis by FINSEQ_4:def 7;
  end;
