reserve X for set,
  x,y,z for Element of BooleLatt X,
  s for set;
reserve y for Element of BooleLatt X;
reserve L for Lattice,
  p,q for Element of L;
reserve A for RelStr,
  a,b,c for Element of A;
reserve A for non empty RelStr,
  a,b,c,c9 for Element of A;
reserve V for with_suprema antisymmetric RelStr,
  u1,u2,u3,u4 for Element of V;
reserve N for with_infima antisymmetric RelStr,
  n1,n2,n3,n4 for Element of N;
reserve K for with_suprema with_infima reflexive antisymmetric RelStr,
  k1,k2,k3 for Element of K;

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;
