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

theorem Th24:
  n >= 1 implies sieve (<%i_FC,1.F_Complex%> `^ (2*n+1),2).(n-1)
    = ((2*n+1) choose 3) * (-i_FC)
proof
A1: i_FC|^1 = i_FC by BINOM:8;
  assume n >=1;
  then reconsider n1=n-1 as Nat;
  n1 <= n1+1 by NAT_1:11;
  then
A2: 2*n1 < 2*n+1 by NAT_1:13,XREAL_1:64;
  then 2*n+1-'(2*n1) = 2*n+1-(2*n1) by XREAL_1:233;
  then
A3: i_FC |^ (2*n+1-'(2*n1)) = i_FC|^(2+1)
    .= (i_FC|^(1+1))* (i_FC|^1) by BINOM:10
    .= i_FC*i_FC*i_FC by A1,BINOM:10
    .=-(1_FC*i_FC) by HAHNBAN1:4,VECTSP_1:9
    .= -i_FC;
A4: (1.FC) |^ (2*n1) = 1_FC by Th3;
  2*n+1-(2*n1) = 3;
  then
A5: (2*n+1) choose (2*n1) = (2*n+1) choose 3 by A2,NEWTON:20;
  sieve (PP `^ (2*n+1),2).n1 = (PP `^ (2*n+1)).(2*n1) by Def5
    .= ((2*n+1) choose 3) * (((1.FC) |^ (2*n1)) * (-i_FC)) by A3,A5,Th13
    .=((2*n+1) choose 3) * (-i_FC) by A4;
  hence thesis;
end;
