reserve a,b,k,m,n,s for Nat;
reserve c,c1,c2,c3 for Complex;
reserve i,j,z for Integer;
reserve p for Prime;
reserve x for object;

theorem
  { n where n is Nat:
    card divisors(3|^(2*n+1),4,1) = card divisors(3|^(2*n+1),4,3)
  } is infinite
  proof
    deffunc A(Nat) = divisors(3|^(2*$1+1),4,1);
    deffunc B(Nat) = divisors(3|^(2*$1+1),4,3);
    set X = {n where n is Nat: card A(n) = card B(n)};
    card A(0) = card B(0) by Th31;
    then
A1: 0 in X;
A2: X is natural-membered
    proof
      let x be object;
      assume x in X;
      then ex n st x = n & card A(n) = card B(n);
      hence thesis;
    end;
    for a st a in X ex b being Nat st b > a & b in X
    proof
      let a;
      assume a in X;
      then consider n such that
A3:   a = n and
      card A(n) = card B(n);
      take n+1;
      thus n+1 > a by A3,NAT_1:16;
      card A(n+1) = card B(n+1) by Th31;
      hence thesis;
    end;
    hence thesis by A1,A2,NUMBER04:1;
  end;
