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

theorem Th12:
  for L being pseudocomplemented distributive bounded Lattice holds
    L is satisfying_Stone_identity iff for a, b being Element of L holds
      (a "/\" b)* = (a*) "\/" (b*)
  proof
    let L be pseudocomplemented distributive bounded Lattice;
    hereby
      assume
 a11: L is satisfying_Stone_identity;
      let a, b be Element of L;
      set g = a "/\" b;
a0:   g "/\" (a* "\/" (b*))
        = ((a*) "/\" g) "\/" (g "/\" (b*)) by LATTICES:def 11
       .= (((a*) "/\" a) "/\" b) "\/" (g "/\" (b*))
         by LATTICES:def 7
       .= (((a*) "/\" a) "/\" b) "\/" (a "/\" (b "/\" (b*)))
         by LATTICES:def 7
       .= ((Bottom L) "/\" b) "\/" (a "/\" (b "/\" (b*))) by ThD
       .= ((Bottom L) "/\" b) "\/" (a "/\" Bottom L) by ThD .= Bottom L;
      for x be Element of L st g "/\" x = Bottom L holds x [= a* "\/" (b*)
      proof
        let x be Element of L;
        set z = b "/\" x;
        set w = x "/\" (a**);
        assume a1: g "/\" x = Bottom L;
a0:     Bottom L [= b "/\" x "/\" (a**) by LATTICES:16;
a2:     (b "/\" x) "/\" a = Bottom L by a1,LATTICES:def 7;
        a* is_a_pseudocomplement_of a by def3; then
        b "/\" x "/\" (a**) [= a* "/\" (a**) by LATTICES:9,a2;
        then b "/\" x "/\" (a**) [= Bottom L by ThD; then
        b "/\" x "/\" (a**) = Bottom L by LATTICES:8,a0; then
a4:     x "/\" (a**) "/\" b = Bottom L by LATTICES:def 7;
        b* is_a_pseudocomplement_of b by def3; then
a5:     x "/\" (a**) [= b* by a4;
a6:     x "/\" (a*) [= a* by LATTICES:6;
        x = x "/\" Top L .= x "/\" ((a*) "\/" (a**)) by a11
        .= (x "/\" (a*)) "\/" (x "/\" (a**)) by LATTICES:def 11;
        hence x [= a* "\/" (b*) by a5,a6,FILTER_0:4;
      end; then
      (a*) "\/" (b*) is_a_pseudocomplement_of (a "/\" b) by a0;
      hence (a "/\" b)* = (a*) "\/" (b*) by def3;
    end;
    assume
a1: for a, b being Element of L holds (a "/\" b)* = (a*) "\/" (b*);
    let x be Element of L;
    x "/\" (x*) = Bottom L by ThD; then
    x* "\/" (x**) = (Bottom L)* by a1;
    hence thesis by Th11;
  end;
