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 :: Problem 38 c
  { n where n is positive Nat: card divisors(n,4,1) > card divisors(n,4,3) }
  is infinite
  proof
    deffunc A(positive Nat) = divisors($1,4,1);
    deffunc B(positive Nat) = divisors($1,4,3);
    set X = { n where n is positive Nat: card A(n) > card B(n) };
    set n = 5|^0;
    {k where k is Nat: k divides n} = A(n) by Th36;
    then
A1: card A(n) = 0+1 by Th37;
    card B(n) = 0 by Th38,CARD_1:27;
    then
A2: n in X by A1;
A3: X is natural-membered
    proof
      let x be object;
      assume x in X;
      then ex n being positive Nat 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 being positive Nat such that
A4:   a = n and card A(n) > card B(n);
      take b = 5|^n;
      thus b > a by A4,NEWTON:86;
A5:   card B(b) = 0 by Th38,CARD_1:27;
      {k where k is Nat: k divides b} = A(b) by Th36;
      then card A(b) = n+1 by Th37;
      hence b in X by A5;
    end;
    hence thesis by A2,A3,NUMBER04:1;
  end;
