 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 L being Stone Lattice, a being Element of L holds
  ex b, c being Element of L st a = b "/\" c & b in Skeleton L &
  c in DenseElements L
  proof
    let L be Stone Lattice, a be Element of L;
A1: a = (a** "/\" a) "\/" Bottom L by LATTICES:4,Th5
     .= (a** "/\" a) "\/" (a** "/\" (a*)) by ThD
     .= a** "/\" (a "\/" (a*)) by LATTICES:def 11;
    take b = a**;
    take c = a "\/" (a*);
G1: (a "\/" (a*))* [= a* by Th6,LATTICES:5;
    (a "\/" (a*))* [= a** by Th6,LATTICES:5; then
    (a "\/" (a*))* [= a* "/\" (a**) by FILTER_0:7,G1; then
    (a "\/" (a*))* [= Bottom L by ThD; then
    (a "\/" (a*))* = Bottom L by BOOLEALG:9;
    hence thesis by A1;
  end;
