
theorem Th31: :: EXAMPLES 4.11.(1b)
  for X be set holds BoolePoset X is algebraic
proof
  let X be set;
  now
    let x be Element of BoolePoset X;
A1: now
      let a be Element of BoolePoset X;
      assume
A2:   a is_>=_than compactbelow x;
      now
        let t be object;
        assume
A3:     t in x;
A4:     x c= X by Th26;
        now
          let k be object;
          assume k in {t};
          then k = t by TARSKI:def 1;
          hence k in X by A3,A4;
        end;
        then {t} c= X;
        then reconsider t1 = {t} as Element of BoolePoset X by Th26;
        for k be object st k in {t} holds k in x by A3,TARSKI:def 1;
        then {t} is finite Subset of x by TARSKI:def 3;
        then {t} in the set of all y where y is finite Subset of x ;
        then {t} in compactbelow x by Th29;
        then t1 <= a by A2,LATTICE3:def 9;
        then t in {t} & {t} c= a by TARSKI:def 1,YELLOW_1:2;
        hence t in a;
      end;
      then x c= a;
      hence x <= a by YELLOW_1:2;
    end;
    for b be Element of BoolePoset X st b in compactbelow x holds b <= x
    by Th4;
    then x is_>=_than compactbelow x by LATTICE3:def 9;
    hence x = sup compactbelow x by A1,YELLOW_0:32;
  end;
  then ( for x be Element of BoolePoset X holds compactbelow x is non empty
  directed)& BoolePoset X is satisfying_axiom_K;
  hence thesis;
end;
