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

theorem Th23:
  sieve (<%i_FC,1.F_Complex%> `^ (2*n+1),2).n = ((2*n+1) choose 1) * i_FC
proof
  2*n+1-'(2*n) = 2*n+1-(2*n) by NAT_1:11,XREAL_1:233;
  then
A1: i_FC |^ (2*n+1-'(2*n)) = i_FC by BINOM:8;
  (1.FC) |^ (2*n) = 1_FC by Th3;
  then
A2: (1.FC) |^ (2*n) * i_FC = i_FC;
  2*n+1-(2*n) = 1;
  then
A3: (2*n+1) choose (2*n) = (2*n+1) choose 1 by NAT_1:11,NEWTON:20;
  sieve (PP `^ (2*n+1),2).n = (PP `^ (2*n+1)).(2*n) by Def5;
  hence thesis by A1,A2,A3,Th13;
end;
