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 b
  { n where n is Nat: card divisors(n,4,1) = card divisors(n,4,3) } is infinite
  proof
    deffunc A(Nat) = divisors($1,4,1);
    deffunc B(Nat) = divisors($1,4,3);
    set A = {n where n is Nat: card A(3|^(2*n+1)) = card B(3|^(2*n+1))};
    set X = {n where n is Nat: card A(n) = card B(n)};
    set n = 3|^(2*0+1);
A1: card A(n) = 1 by Th29;
    card B(n) = 1 by Th30;
    then
A2: n in X by A1;
A3: 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;
      take b = 3|^(2*a+1);
      1*a <= 2*a by XREAL_1:64;
      then
A4:   a+0 < 2*a+1 by XREAL_1:8;
      b > 2*a+1 by NEWTON:86;
      hence b > a by A4,XXREAL_0:2;
A5:   card A(b) = a+1 by Th29;
      card B(b) = a+1 by Th30;
      hence thesis by A5;
    end;
    hence thesis by A2,A3,NUMBER04:1;
  end;
