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

theorem
  for L being pseudocomplemented distributive bounded Lattice holds
    L is satisfying_Stone_identity iff for a, b being Element of L st
    a in Skeleton L & b in Skeleton L holds a "\/" b in Skeleton L
  proof
    let L be pseudocomplemented distributive bounded Lattice;
    hereby
      assume
  a0: L is satisfying_Stone_identity;
      let a, b be Element of L;
      assume
X0:   a in Skeleton L & b in Skeleton L; then
      consider x being Element of L such that
X1:   a = x*;
      consider y being Element of L such that
X2:   b = y* by X0;
      (x "/\" y)* in Skeleton L;
      hence a "\/" b in Skeleton L by X1,X2,Th12,a0;
    end;
    assume
a1: for a, b being Element of L st a in Skeleton L & b in Skeleton L
    holds a "\/" b in Skeleton L;
    let y be Element of L;
    y* in Skeleton L & y** in Skeleton L; then
sp: (y* "\/" (y**))** = y* "\/" (y**) by Th13,a1;
r1: y** [= (y* "/\" (y**))* by Th6,LATTICES:6;
    y*** [= (y* "/\" (y**))* by LATTICES:6,Th6; then
    y* [= (y* "/\" (y**))* by Th7; then
r:  (y*) "\/" (y**) [= (y* "/\" (y**))* "\/" ((y* "/\" (y**))*)
      by r1,FILTER_0:4;
s1: (y* "\/" (y**))* [= y** by Th6,LATTICES:5;
    (y* "\/" (y**))* [= y*** by Th6,LATTICES:5; then
s2: (y* "\/" (y**))* [= y* by Th7;
    (y* "\/" (y**))* "/\" ((y* "\/" (y**))*) [= y* "/\" (y**)
      by s1,s2,FILTER_0:5; then
    (y* "/\" (y**))* [= y* "\/" (y**) by sp,Th6;then
    (y* "/\" (y**))* = y* "\/" (y**) by r,LATTICES:8;
    then (Bottom L)* = y* "\/" (y**) by ThD;
    hence thesis by Th11;
  end;
