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

theorem Th32:
  for X be non empty set,F be non empty Subset of BooleLatt X holds
  F is Filter of BooleLatt X iff
  for Y1,Y2 be Subset of X holds
  (Y1 in F & Y2 in F implies Y1/\Y2 in F) &
  (Y1 in F & Y1 c= Y2 implies Y2 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;
     hereby
       let Y1,Y2 be Subset of X;
       Y1 is Element of the carrier of BooleLatt X & Y2 is
       Element of the carrier of BooleLatt X by LATTICE3:def 1;
       hence (Y1 in F & Y2 in F implies Y1/\Y2 in F) &
       (Y1 in F & Y1 c= Y2 implies Y2 in F) by A1,Th31;
     end;
   end;
   assume that
A2: for Y1,Y2 be Subset of X holds
    (Y1 in F & Y2 in F implies Y1/\Y2 in F) &
    (Y1 in F & Y1 c= Y2 implies Y2 in F);
    now
      hereby
        let p,q be Element of F;
        reconsider p1=p,q1= q as Subset of X by LATTICE3:def 1;
        p1 in F & q1 in F implies p1/\q1 in F by A2;
        hence p/\q in F;
      end;
      let p be Element of F,q being Element of BooleLatt X such that
A3:   p c= q;
      reconsider p1=p,q1=q as Subset of X by LATTICE3:def 1;
      p1 in F & p1 c= q &
      (p1 in F & p1 c= q1 implies q1 in F) by A2,A3;
      hence q in F;
    end;
    hence  F is Filter of BooleLatt X by Th31;
  end;
