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(5|^n,4,1) > card divisors(5|^n,4,3) }
  is infinite
  proof
    deffunc A(Nat) = divisors(5|^$1,4,1);
    deffunc B(Nat) = divisors(5|^$1,4,3);
    set X = {n where n is Nat: card A(n) > card B(n)};
    {k where k is Nat: k divides 5|^0} = A(0) by Th36;
    then card A(0) = 0+1 by Th37;
    then card A(0) > card B(0) by Th38,CARD_1:27;
    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 where a is Nat: a divides 5|^(n+1)} = n+1+1 by Th37;
      then card A(n+1) = n+1+1 by Th36;
      then card A(n+1) > card B(n+1) by Th38,CARD_1:27;
      hence thesis;
    end;
    hence thesis by A1,A2,NUMBER04:1;
  end;
