theorem Th25:
  BooleLatt X is complete
proof
  set B = BooleLatt X;
  let x be set;
  set p = union (x /\ bool X);
A1: carr(B) = bool X by Def1;
  reconsider p as Element of B by Def1;
  take p;
  thus x is_less_than p
  proof
    let q be Element of B;
    assume q in x;
    then q in x /\ bool X by A1,XBOOLE_0:def 4;
    then q c= p by ZFMISC_1:74;
    hence thesis by Th2;
  end;
  let r be Element of B such that
A2: for q being Element of B st q in x holds q [= r;
  now
    let z be set;
    assume
A3: z in x /\ bool X;
    then
A4: z in x by XBOOLE_0:def 4;
    reconsider z9 = z as Element of B by A1,A3;
    z9 [= r by A2,A4;
    hence z c= r by Th2;
  end;
  then p c= r by ZFMISC_1:76;
  hence thesis by Th2;
end;
