 reserve R for domRing;
 reserve p for odd prime Nat, m for positive Nat;
 reserve g for non zero Polynomial of INT.Ring;

theorem Th12:
  for p be odd prime Nat, m be positive Nat holds
  len ~(f_0(m,p)) = m*p + p
    proof
      let p be odd prime Nat, m be positive Nat;
      set xp0 = x.(m,p);
      set pp0 = (Product(x.(m,p))), t0 = (tau(0))|^(p-'1);
A1:   len ~(Product (x.(m,p))) = m*p +1 by Th11;
A2:   p-'1 = p-1 by XREAL_1:233, NAT_1:14;
      len ~((tau(0))|^(p-'1)) = p-'1+1 by Th8; then
A4:   (len ~pp0)*(len ~t0) <> 0 by A1;
A5:   len ~pp0 = m*p + 1 by Th11;
      len ~t0 = p-'1 + 1 by Th8; then
      len ~(pp0*t0) = m*p +1 + p-1 by A2,A5,A4,Th9;
      hence thesis by GROUP_4:6;
    end;
