theorem Th31:
  card divisors(3|^(2*n+1),4,1) = card divisors(3|^(2*n+1),4,3)
  proof
    deffunc A(Nat) = divisors(3|^(2*n+1),4,1);
    deffunc B(Nat) = divisors(3|^(2*n+1),4,3);
    deffunc A1(Nat) = { 3|^m: m is even & m <= 2*$1+1 };
    deffunc B1(Nat) = { 3|^m: m is odd & m <= 2*$1+1 };
A1: A(n) = A1(n) by Th25;
A2: B(n) = B1(n) by Th26;
    card A1(n) = n+1 by Th27;
    hence thesis by A1,A2,Th28;
  end;
