
theorem
  for X be set for F be Subset of X holds F in the carrier of
  CompactSublatt BoolePoset X iff F is finite
proof
  let X be set;
  let F be Subset of X;
  reconsider F9 = F as Element of BoolePoset X by Th26;
A1: F c= F;
  thus F in the carrier of CompactSublatt BoolePoset X implies F is finite
  proof
    assume
A2: F in the carrier of CompactSublatt BoolePoset X;
    the carrier of CompactSublatt BoolePoset X c= the carrier of
    BoolePoset X by YELLOW_0:def 13;
    then reconsider F9 = F as Element of BoolePoset X by A2;
    F9 <= F9 & F9 is compact by A2,Def1;
    then F9 in compactbelow F9;
    then F9 in the set of all y where y is finite Subset of F9  by Th29;
    then ex F1 be finite Subset of F9 st F1 = F9;
    hence thesis;
  end;
  assume F is finite;
  then F in the set of all y where y is finite Subset of F9  by A1;
  then F9 in compactbelow F9 by Th29;
  then F9 is compact by Th4;
  hence thesis by Def1;
end;
