 reserve n,i for Nat;
 reserve p for Prime;

theorem
  for n being non zero Nat holds
    Divisors_Lattice n is Boolean iff n is square-free
  proof
    let n be non zero Nat;
    set L = Divisors_Lattice n;
    thus L is Boolean implies n is square-free
    proof
      assume
A0:   L is Boolean;
      assume n is square-containing; then
      consider p being Prime such that
A1:   p |^ 2 divides n by MOEBIUS1:def 1;
a1:   p * p divides n by A1,NEWTON:81;
A2:   p <> 1 by INT_2:def 4;
      p divides p * p; then
      p divides p |^ 2 by NEWTON:81; then
      p in NatDivisors n by MOEBIUS1:39,A1,NAT_D:4; then
      reconsider pp = p as Element of L by DivLat;
      not ex a being Element of L st a is_a_complement_of pp
      proof
        given a being Element of L such that
C1:     a is_a_complement_of pp;
C2:     a lcm p = n & a gcd p = 1 by C1,TopBot;
        a, p are_coprime by C1,TopBot; then
        a lcm p = a * p by INT615; then
        p divides a by C2,a1,INT_4:7; then
        p divides a gcd p by NAT_D:def 5; then
C5:     p <= a gcd p by NAT_D:7;
        a gcd p <> 1
        proof
          assume a gcd p = 1; then
          p < 1 + 1 by C5,NAT_1:13;
          hence thesis by A2,NAT_1:23;
        end;
        hence thesis by C1,TopBot;
      end;
      hence thesis by A0,LATTICES:def 19;
    end;
    thus thesis;
  end;
