reserve
  X for non empty set,
  FX for Filter of X,
  SFX for Subset-Family of X;

theorem Th31:
  for X be non empty set, F being non empty Subset of BooleLatt X holds
  F is Filter of BooleLatt X iff
  (for p,q being Element of F holds p/\q in F) &
  (for p being Element of F, q be Element of BooleLatt X st
  p c= q holds q in F)
  proof
    let X be non empty set,F be non empty Subset of BooleLatt X;
    hereby
      assume
A1:   F is Filter of BooleLatt X; then
A2:   for p,q being Element of BooleLatt X st p in F & p [= q holds
      q in F by FILTER_0:9;
      now
        let p,q be Element of BooleLatt X;
        now
          assume p in F & q in F;
          then p"/\"q in F by A1,FILTER_0:9;
          hence p/\q in F;
        end;
        hence (p in F & q in F implies p/\q in F) &
        (p in F & p c= q implies q in F) by A2,LATTICE3:2;
      end;
      hence (for p,q being Element of F holds p/\q in F) &
      (for p be Element of F,q be Element of BooleLatt X st
      p c= q holds q in F);
    end;
    assume that
A3: for p,q being Element of F holds p/\q in F and
A4: for p be Element of F, q be Element of BooleLatt X st p c= q holds q in F;
A5: (for p,q being Element of BooleLatt X st p in F & q in F holds
    p"/\"q in F) by A3;
    for p,q being Element of BooleLatt X st p in F & p [= q holds
    q in F by A4,LATTICE3:2;
    hence F is Filter of BooleLatt X by A5,FILTER_0:9;
  end;
