 reserve L for Lattice;
 reserve I,P for non empty ClosedSubset of L;
reserve L for lower-bounded pseudocomplemented Lattice;
reserve L for Stone Lattice;

theorem
  for p being Prime holds NatDivisors p = {1, p}
  proof
    let p be Prime;
a1: NatDivisors (p|^1)={ p|^k where k is Element of NAT:k <= 1 } by NAT_5:19;
    { p|^k where k is Element of NAT : k <= 1 } = {1, p}
    proof
      thus { p|^k where k is Element of NAT : k <= 1 } c= {1, p}
      proof
        let x be object;
        assume x in { p|^k where k is Element of NAT : k <= 1 }; then
        consider kk being Element of NAT such that
A2:     x = p |^ kk & kk <= 1;
        kk = 0 or ... or kk = 1 by A2; then
        x = 1 or x = p |^ 1 by NEWTON:4,A2;
        hence thesis by TARSKI:def 2;
      end;
      let x be object;
      assume x in {1, p}; then
      x = 1 or x = p by TARSKI:def 2; then
      x = p |^ 0 or x = p |^ 1 by NEWTON:4;
      hence thesis;
    end;
    hence thesis by a1;
  end;
