
theorem Th11:
  for X being set, Y being non empty upper Subset of BoolePoset X
  holds Y is filtered iff for Z being finite Subset-Family of X st Z c= Y holds
  Intersect Z in Y
proof
  let X be set, Y be non empty upper Subset of BoolePoset X;
  hereby
    assume
A1: Y is filtered;
    then Top BoolePoset X in Y by WAYBEL_4:22;
    then
A2: X in Y by YELLOW_1:19;
    let Z be finite Subset-Family of X;
    reconsider B = Z as Subset of BoolePoset X by Th2;
    assume Z c= Y;
    then reconsider A = Z as finite Subset of Y;
    A <> {} implies inf B in Y & inf B = meet B by A1,WAYBEL_0:43,YELLOW_1:20;
    hence Intersect Z in Y by A2,SETFAM_1:def 9;
  end;
  assume
A3: for Z being finite Subset-Family of X st Z c= Y holds Intersect Z in Y;
A4: the carrier of BoolePoset X = bool X by Th2;
  now
    let A be finite Subset of Y;
    reconsider Z = A as finite Subset-Family of X by A4,XBOOLE_1:1;
    assume
A5: A <> {};
    reconsider Z as finite Subset-Family of X;
    A c= the carrier of BoolePoset X by XBOOLE_1:1;
    then "/\"(A, BoolePoset X) = meet Z by A5,YELLOW_1:20
      .= Intersect Z by A5,SETFAM_1:def 9;
    hence "/\"(A, BoolePoset X) in Y by A3;
  end;
  hence thesis by WAYBEL_0:43;
end;
