 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 PsCompl:
  for L being Lattice, p being Prime, x being Element of L st
    L = Divisors_Lattice (p * p) & x = p holds x* = Bottom L
  proof
    let L be Lattice, p be Prime, x be Element of L;
    assume that
A1: L = Divisors_Lattice (p * p) and
B0: x = p;
    reconsider pp = Bottom L as Element of L;
    for y being Element of L st x "/\" y = Bottom L holds y [= pp
    proof
      let y be Element of L;
      y in the carrier of L; then
      y in NatDivisors (p*p) by A1,MOEBIUS2:def 10; then
      y in {1, p, p*p} by DivisorsSquare; then
w3:   y = 1 or y = p or y = p * p by ENUMSET1:def 1;
      assume
W1:   x "/\" y = Bottom L;
W2:   y <> Top L
      proof
        assume y = Top L; then
        x = 1 by MOEBIUS2:64,A1,W1;
        hence thesis by INT_2:def 4,B0;
      end;
      y <> p
      proof
        assume y = p; then
        1 = p by A1,MOEBIUS2:64,B0,W1;
        hence thesis by INT_2:def 4;
      end;
      hence thesis by W2,w3,A1,MOEBIUS2:64;
    end; then
    pp is_a_pseudocomplement_of x by A1;
    hence thesis by def3,A1;
  end;
