reserve L for Lattice,
  p,p1,q,q1,r,r1 for Element of L;
reserve x,y,z,X,Y,Z,X1,X2 for set;

theorem Th8:
 for S being non empty Subset of L holds S is Filter of L iff
  for p,q being Element of L holds p in S & q in S iff p "/\" q in S
 proof let S be non empty Subset of L;
  thus S is Filter of L implies
  for p,q being Element of L holds p in S & q in S iff p "/\" q in S
     by LATTICES:6,def 23,def 24;
  assume
A1:  for p,q being Element of L holds p in S & q in S iff p "/\" q in S;
    S is final proof let p,q be Element of L such that
A2:    p [= q and
A3:    p in S;
       p "/\" q = p by A2,LATTICES:4;
      hence q in S by A1,A3;
     end;
  hence thesis by A1,LATTICES:def 24;
 end;
