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

theorem Th25:
  len sieve (<%i_FC,1.F_Complex%> `^ (2*n+1),2) = n + 1
proof
  set PPn=PP`^(2*n+1);
A1:n+1 is_at_least_length_of sieve (PPn,2)
  proof
    let i be Nat;
    assume i >= n+1;
    then i+i >= n+1+(n+1) & 2*n+1+1 > 2*n+1+0 by XREAL_1:7,8;
    then 2*i > 2*n+1 by XXREAL_0:2;
    then
A2: (2*n+1) choose (2*i) = 0 by NEWTON:def 3;
    thus sieve (PPn,2).i = PPn.(2*i) by Def5
      .= 0 * ((1.FC) |^ (2*i) * (i_FC|^ (2*n+1-'(2*i)))) by A2,Th13
      .= 0.FC;
  end;
  for m st m is_at_least_length_of sieve (PPn,2) holds n+1 <=m
  proof
    let m such that
A3: m is_at_least_length_of sieve (PPn,2);
    assume m < n+1;
    then m <= n by NAT_1:13;
    then sieve(PPn,2).n = 0.FC by A3,ALGSEQ_1:def 2;
    then ((2*n+1) choose 1) * i_FC = 0.FC by Th23;
    hence thesis;
  end;
  hence thesis by A1,ALGSEQ_1:def 3;
end;
