
theorem
  for X be set for a be Element of BoolePoset X holds uparrow a = {Y
  where Y is Subset of X : a c= Y}
proof
  let X be set;
  let a be Element of BoolePoset X;
A1: {Y where Y is Subset of X : a c= Y} c= uparrow a
  proof
    let x be object;
    assume x in {Y where Y is Subset of X : a c= Y};
    then consider x9 be Subset of X such that
A2: x9 = x and
A3: a c= x9;
    reconsider x9 as Element of BoolePoset X by WAYBEL_8:26;
    a <= x9 by A3,YELLOW_1:2;
    hence thesis by A2,WAYBEL_0:18;
  end;
  uparrow a c= {Y where Y is Subset of X : a c= Y}
  proof
    let x be object;
    assume
A4: x in uparrow a;
    then reconsider x9 = x as Element of BoolePoset X;
    a <= x9 by A4,WAYBEL_0:18;
    then x9 is Subset of X & a c= x9 by WAYBEL_8:26,YELLOW_1:2;
    hence thesis;
  end;
  hence thesis by A1,XBOOLE_0:def 10;
end;
