reserve k,m,n for Nat;
reserve R for commutative Ring,
        p,q for Polynomial of R,
        z0,z1 for Element of R;

theorem Th19:
  for L being non empty ZeroStr
    for p being sequence of L holds
      sieve(p,2*k) = sieve(even_part p,2*k)
  proof
    let L be non empty ZeroStr;
    let p be sequence of L;
    let n be Element of NAT;
    thus (sieve (even_part p,2*k)).n = (even_part p).(2*k*n) by Def5
    .= p.(2*k*n) by HURWITZ2:def 1
    .=(sieve (p,2*k)).n by Def5;
  end;
