 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;
 reserve L1, L2 for Lattice;
 reserve p1, q1 for Element of L1;
 reserve p2, q2 for Element of L2;
 reserve L1, L2 for non empty Lattice;
reserve B for Boolean Lattice;

theorem
  DenseElements (B squared-latt) = the set of all [a,Top B]
    where a is Element of B
  proof
    set L = B squared-latt;
    thus DenseElements L c= the set of all [a,Top B] where a is Element of B
    proof
      let x be object;
      assume x in DenseElements L; then
      consider a being Element of L such that
A1:   x = a & a* = Bottom L;
      x in the carrier of L by A1; then
      x in B squared by SquaredCarrier; then
      consider a1, a2 being Element of B such that
A2:   x = [a1,a2] & a1 [= a2;
A3:   a* = [Bottom B, Bottom B] by A1,SquaredBottom;
      a* = [a2`,a2`] by A1,A2,PseudoInSquared; then
      a2`` = (Bottom B)` by A3,XTUPLE_0:1; then
      a2 = Top B by LATTICE4:30;
      hence thesis by A2;
    end;
    let x be object;
    assume x in the set of all [a,Top B] where a is Element of B; then
    consider a being Element of B such that
A1: x = [a, Top B];
    a [= Top B by LATTICES:19; then
    x in B squared by A1; then
    reconsider y = x as Element of L by SquaredCarrier;
    y* = [(Top B)`,(Top B)`] by A1,PseudoInSquared; then
    y* = [Bottom B, (Top B)`] by LATTICE4:29
     .= [Bottom B, Bottom B] by LATTICE4:29
     .= Bottom L by SquaredBottom;
    hence thesis;
  end;
