
theorem Th27:
  for X be set for x,y be Element of BoolePoset X holds x << y iff
for Y be Subset-Family of X st y c= union Y ex Z be finite Subset of Y st x c=
  union Z
proof
  let X be set;
  let x,y be Element of BoolePoset X;
  LattPOSet BooleLatt X = RelStr(#the carrier of BooleLatt X, LattRel (
    BooleLatt X)#) by LATTICE3:def 2;
  then
A1: the carrier of BoolePoset X = the carrier of BooleLatt X by YELLOW_1:def 2
    .= bool X by LATTICE3:def 1;
  thus x << y implies for Y be Subset-Family of X st y c= union Y ex Z be
  finite Subset of Y st x c= union Z
  proof
    assume
A2: x << y;
    let Y be Subset-Family of X;
    reconsider Y9 = Y as Subset of BoolePoset X by A1;
    assume y c= union Y;
    then y c= sup Y9 by YELLOW_1:21;
    then y <= sup Y9 by YELLOW_1:2;
    then consider Z be finite Subset of BoolePoset X such that
A3: Z c= Y and
A4: x <= sup Z by A2,WAYBEL_3:18;
    reconsider Z9 = Z as finite Subset of Y by A3;
    take Z9;
    x c= sup Z by A4,YELLOW_1:2;
    hence thesis by YELLOW_1:21;
  end;
  thus (for Y be Subset-Family of X st y c= union Y ex Z be finite Subset of Y
  st x c= union Z) implies x << y
  proof
    assume
A5: for Y be Subset-Family of X st y c= union Y ex Z be finite Subset
    of Y st x c= union Z;
    now
      let Y be Subset of BoolePoset X;
      reconsider Y9 = Y as Subset-Family of X by A1;
      assume y <= sup Y;
      then y c= sup Y by YELLOW_1:2;
      then y c= union Y9 by YELLOW_1:21;
      then consider Z9 be finite Subset of Y9 such that
A6:   x c= union Z9 by A5;
      reconsider Z = Z9 as finite Subset of BoolePoset X by XBOOLE_1:1;
      take Z;
      thus Z c= Y;
      x c= sup Z by A6,YELLOW_1:21;
      hence x <= sup Z by YELLOW_1:2;
    end;
    hence thesis by WAYBEL_3:19;
  end;
end;
