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

theorem Th21:
  for L being non empty ZeroStr
    for p being Polynomial of L st len even_part p = 0
       for n be non zero Nat holds
      len sieve(p,2*n) = 0
  proof
    let L be non empty ZeroStr;
    let p be Polynomial of L such that A1: len even_part p =0;
    let n be non zero Nat;
    A2:0 is_at_least_length_of even_part p by A1,ALGSEQ_1:def 3;
    0 is_at_least_length_of sieve (p,2*n)
    proof
      let k such that k >=0;
      thus sieve (p,2*n).k  = p.(2*n*k) by Def5
      .= (even_part p).(2*n*k) by HURWITZ2:def 1
      .= 0.L by A2,ALGSEQ_1:def 2;
    end;
    hence thesis by ALGSEQ_1:def 3;
  end;
