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

theorem Th27:
  Roots (sieve (<%i_FC,1.F_Complex%> `^ (2*n+1),2)) = rng sqr cot x_r-seq(n)
proof
  set f=x_r-seq(n),F=sqr cot f,C=sieve(PP `^ (2*n+1),2);
A1:len F = len cot f = len f =n by CARD_1:def 7,BASEL_1:21;
  F is one-to-one by BASEL_1:28;
  then
A2:card dom F = card rng F & dom F = Seg n by A1,CARD_1:70,FINSEQ_1:def 3;
A3:rng F c= Roots C by Th26;
A4:n <= card Roots C by A2,NAT_1:43,Th26;
  card Roots C < len C by Th11;
  then card Roots C < n+1 by Th25;
  then card Roots C <= n by NAT_1:13;
  hence thesis by A3,CARD_2:102,A2,A4,XXREAL_0:1;
end;
